RD Sharma solutions for Mathematics [English] Class 12 chapter 5 - Algebra of Matrices [2018 edition]

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

If A = [a_ij] =`[[2,3,-5],[1,4,9],[0,7,-2]]`and B = [b_ij] `[[2,-1],[-3,4],[1,2]]`

Let A be a matrix of order 3 × 4. If R₁ denotes the first row of A and C₂ denotes its second column, then determine the orders of matrices R₁ and C₂

Construct a 2 × 3 matrix whose elements a_ij are given by :

(i) a_ij = i . j

Construct a 2 × 3 matrix whose elements a_ij are given by :

(ii) a_ij = 2i − j

Construct a 2 × 3 matrix whose elements a_ij are given by :

(iii) a_ij = i + j

Construct a 2 × 3 matrix whose elements a_ij are given by :

(iv) a_ij =`(i+j)^2/2` 

Construct a 2 × 2  matrix whose elements `a_(ij)`

are given by: `(i+j)^2/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`aij=(i-j)^2/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`a_(ij)=(i-2_j)^2/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`a_(ij)= (2i +j)^2/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`a_(ij)=|2_i - 3_i|/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`a_(ij)=|-3i +j|/2`

Construct a 2 × 2 matrix whose elements a_ij are given by:

`a_(ij)=e^(2ix) sin (xj)`

Construct a 3 × 4 matrix A = [a_ij] whose elements a_ij are given by:

a_ij = i + j

Construct a 3 × 4 matrix A = [a_jj] whose elements a_jj are given by:

a_jj = i − j

Construct a 3 × 4 matrix A = [a_ij] whose elements a_ij are given by:

 a_ij = 2i

Construct a 3 × 4 matrix A = [a_ij] whose elements a_ij are given by:

a_ij = j

Construct a 3 × 4 matrix A = [a_ij] whose elements a_ij are given by:

`a_(ij)=1/2= -3i + j `

Construct a 4 × 3 matrix whose elements are

`a_(ij)=2_i+ i/j`

Construct a 4 × 3 matrix whose elements are

`a_(ij)= (i-j)/(i+j )`

Construct a 4 × 3 matrix whose elements are

 a_ij = i 

find x, y , a and b if  \[\begin{bmatrix}3x + 4y & 2 & x - 2y \\ a + b & 2a - b & - 1\end{bmatrix} = \begin{bmatrix}2 & 2 & 4 \\ 5 & - 5 & - 1\end{bmatrix}\] 

Find x, y, a and b if

`[[2x-3y,a-b,3],[1,x+4y,3a+4b]]`=`[[1,-2,3],[1,6,29]]`

Find the values of a, b, c and d from the following equations:`[[2a + b,a-2b],[5c-d,4c + 3d ]]`= `[[4,- 3],[11,24]]`

Find x, y and z so that A = B, where`A= [[x-2,3,2x],[18z,y+2,6x]],``b=[[y,z,6],[6y,x,2y]]`

`If [[x,3x- y],[2x+z,3y -w ]]=[[3,2],[4,7]]` find x,y,z,w

`If [[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]`Find X,Y,Z,W.

`If [[x + 3 , z + 4 ,     2y-7 ],[4x + 6,a-1,0 ],[b-3,3b,z + 2c ]]= [[0,6,3y-2],[2x,-3,2c-2],[2b + 4,-21,0]]`Obtain the values of a, b, c, x, y and z.

`If [[2x +1    5x],[0     y^2 +1]]``= [[x+3   10],[0      26 ]]`, find the value of (x + y).

`If [[xy          4],[z+6     x+y ]]``=[[8     w],[0     6]]`, then find the values of X,Y,Z and W . 

Given an example of

a row matrix which is also a column matrix,

Given an example of

a diagonal matrix which is not scalar,

Given an example of

 a triangular matrix

The sales figure of two car dealers during January 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars. Write 2 × 3 matrices summarizing sales data for January and 2-month period for each dealer.

For what values of x and y are the following matrices equal?

`A=[[2x+1   2y],[0              y^2 - 5y]]``B=[[x + 3      y^2 +2],[0        -6]]`

Find the values of x and y if

`[[X + 10,Y^2 + 2Y],[0, -4]]`=`[[3x +4,3],[0,y^2-5y]]`

For what values of a and b if A = B, where

`A = [[a + 4        3b],[8        -6]]   B = [[2a +2              b^2+2],[8                    b^2  - 5b]]`

Disclaimer: There is a misprint in the question, b² − 5b should be written instead of b² − 56.

Compute the following sums:

`[[3   -2],[1           4]]+ [[-2         4 ],[1           3]]`

Compute the following sums:

`[[2    1   3],[0   3   5],[-1   2   5]]`+ `[[1 -2     3],[2            6        1],[0   -3       1]]`

Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: 2A − 3B

Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following:  B − 4C

Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: 3A − C

Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: 3A − 2B + 3C

If A =`[[2,3],[5,7]],B =` `[[-1,0 ,2],[3,4,1]]`,`C= [[-1,2,3],[2,1,0]]`find : A + B and B + C

If A =`[[2   3],[5   7]],B =` `[[-1   0   2],[3    4      1]]`,`C= [[-1    2   3],[2    1     0]]`find

2B + 3A and 3C − 4B

Let A = `[[-1    0    2],[3     1      4]]``B=[[0      -2     5],[1      -3     1]]``and C = [[1     -5       2],[6     0    -4 ]]`Compute2A2-3B +4C : 

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find: A − 2B

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find

B + C − 2A

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find

2A + 3B − 5C

Given the matrices 

`A=[[2,1,1],[3,-1,0],[0,2,4]]` , `B=[[9,7,-1],[3,5,4],[2,1,6]]`  `and  C=[[2,-4,3],[1,-1,0],[9,4,5]]`

Find matrices X and Y, if X + Y =`[[5     2],[0       9]]`

and X − Y =  `[[3       6],[0   -1]]`

Find X if Y =`[[3       2],[1      4]]`and 2X + Y =`[[1       0],[-3        2]]`

Find matrices X and Y, if 2X − Y = `[[6       -6           0],[-4            2           1]]`and X + 2Y =`[[3              2                     5],[-2         1    -7 ]]`

f X − Y =`[[1      1       1],[1        1          0],[1         0          0]]` and X + Y = `[[3        5         1],[-1       1           1],[11       8           0]]`find X and Y.

Find matrix A, if  `[[1         2      -1],[0         4       9]]`

`+ A = [[9        -1           4],[-2        1            3]]`

If A =`[[9     1],[7      8]],B=[[1      5],[7      12]]`find matrix C such that 5A + 3B + 2C is a null matrix.

If A = `[[2      -2],[4             2],[-5          1]],B=[[8             0],[4      -2],[3          6]]`

If A = `[[1    -3         2],[2        0               2]]`and `B = [[2          -1           -1],[1           0             -1]]` find the matrix C such that A + B + C is 

Find x, y satisfying the matrix equations

`[[X-Y               2            -2],[4                        x                6]]+[[3        -2                2],[1         0            -1]]=[[                6                       0                             0],[         5                       2x+y                5]]`

Find x, y satisfying the matrix equations

`[x     y + 2    z-3 ] +  [  y       4          5]=[4        9        12]`

Find x, y satisfying the matrix equations

`x[[2],[1]]+y[[3],[5]]+[[-8],[-11]]=0`

If 2 `[[3    4],[5     x]]+[[1   y],[0    1]]=[[7        0],[10      5]]` find x and y.

Find the value of λ, a non-zero scalar, if λ

`[[1      0         2],[3        4         5]]+2[[1              2             3],[-1   -3     2]]`=`[[4     4     10],[4      2       14]]`

Find a matrix X such that 2A + B + X = O, where

`A= [[-1      2],[3        4]],B= [[3       -2],[1          5]]`

Find a matrix X such that 2A + B + X = O, where 

 If A = `[[8            0],[4    -2],[3         6]]` and B = `[[2       -2],[4           2],[-5          1]]`

Find x, y, z and t, if

`3[[x     y],[z      t]]=[[x        6],[-1          2t]]+[[4             x+y],[z+t         3]]`

Find x, y, z and t, if

`2[[x         5],[z         t]]+[[x           6],[-1          2t]]=[[7            14],[15        14]]`

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

`2X + 3Y = [[2,3],[4,0]], 3X+2Y = [[-2,2],[1,-5]]`

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Compute the indicated products:

`[[a    b],[-b      a]][[a     -b],[b         a]]`

Compute the indicated products:

`[[1     -2],[2     3]][[1         2        3],[-3    2      -1]]`

Compute the indicated product:

`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`

Show that AB ≠ BA in each of the following cases:

`A= [[5    -1],[6        7]]`And B =`[[2       1],[3         4]]`

Show that AB ≠ BA in each of the following cases

`A=[[-1          1           0],[0          -1           1],[2                  3                4]]`  and  =B `[[1          2            3], [0          1           0],[1        1          0]]`

Show that AB ≠ BA in each of the following cases:

`A=[[1       3         0],[1        1          0],[4         1         0]]`And    B=`[[0      1          0],[1        0        0],[0           5          1]]`

Compute the products AB and BA whichever exists in each of the following cases:

`A= [[1      -2],[2              3]]` and  B=`[[1       2        3],[2         3         1]]`

Compute the products AB and BA whichever exists in each of the following cases:

`A=[[3     2],[-1     0],[-1      1]]` and `B= [[4         5        6],[0           1             2]]`

Compute the products AB and BA whichever exists in each of the following cases:

A = [1 −1 2 3] and B=`[[0],[1],[3],[2]]`

Compute the products AB and BA whichever exists in each of the following cases:

 [a, b]`[[c],[d]]`+ [a, b, c, d] `[[a],[b],[c],[d]]`

Show that AB ≠ BA in each of the following cases:

`A = [[1,3,-1],[2,-1,-1],[3,0,-1]]` And `B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]`

Evaluate the following:

`([[1              3],[-1    -4]]+[[3        -2],[-1         1]])[[1         3           5],[2            4               6]]`

Evaluate the following:

`[[],[1  2  3],[]]` `[[1     0      2],[2       0         1],[0          1       2]]` `[[2],[4],[6]]`

Evaluate the following:

`[[1     -1],[0            2],[2           3]]`  `([[1     0        2],[2        0        1]]-[[0             1                 2],[1           0                    2]])`

If A = `[[1     0],[0        1]]`,B`[[1            0],[0       -1]]`

and C= `[[0      1],[1       0]]` 

If A = `[[2       -1],[3             2]]`  and B = `[[0         4],[-1          7]]`find 3A² − 2B + I

If A =  `[[4       2],[-1        1]]` 

If A =  `[[1    1],[0    1]]`  show that A² = `[[1       2],[0          1]]` and A³ = `[[1        3],[0       1]]`

If A = `[[ab,b^2],[-a^2,-ab]]` , show that A² = O

If A = `[[ cos 2θ     sin 2θ],[ -sin 2θ    cos 2θ]]`, find A².

If A =

\[\begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\]and B =

\[\begin{bmatrix}- 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5\end{bmatrix}\] , show that AB = BA = O_3×3.

If A = `[[0,c,-b],[-c,0,a],[b,-a,0]]`and B =`[[a^2 ,ab,ac],[ab,b^2,bc],[ac,bc,c^2]]`, show that AB = BA = O_3×3.

If A =`[[2     -3          -5],[-1             4           5],[1           -3       -4]]` and B =`[[2         -2            -4],[-1               3                  4],[1            2           -3]]`

Let A =`[[-1            1               -1],[3         -3           3],[5           5             5]]`and B =`[[0                4                  3],[1              -3              -3],[-1               4                 4]]`

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):

`A =-[[1             2         0],[-1        0           1]]`,`B=[[1       0],[-1        2],[0        3]]` and C= `[[1],[-1]]`

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):

`A=[[4       2        3],[1       1          2],[3         0          1]]`=`B=[[1        -1          1],[0         1            2],[2           -1          1]]` and  `C= [[1       2       -1],[3       0         1],[0         0         1]]` 

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:

`A = [[1     -1],[0          2]] B=   [[-1       0],[2        1]]`and `C= [[0       1],[1     -1]]`

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:

`A=[[2    -1],[1        1],[-1         2]]` `B=[[0     1],[1      1]]` C=`[[1      -1],[0                1]]`

If A= `[[1        0           -2],[3        -1           0],[-2              1               1]]` B=,`[[0         5           -4],[-2          1             3],[-1          0              2]] and  C=[[1               5              2],[-1           1              0],[0          -1             1]]` verify that A (B − C) = AB − AC.

Compute the elements a₄₃ and a₂₂ of the matrix:`A=[[0     1        0],[2      0        2],[0       3        2],[4        0       4]]` `[[2       -1],[-3           2],[4              3]]  [[0            1           -1                    2                     -2],[3       -3             4          -4                  0]]`

If w is a complex cube root of unity, show that

`([[1         w          w^2],[w            w^2             1],[w^2           1             w]]+[[w          w^2          1],[w^2             1               w],[w            w^2              1]])[[1],[w],[w^2]]=[[0],[0],[0]]`

\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\]   , Show that A² = A.

 If  \[A = \begin{bmatrix}4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3\end{bmatrix}\]     ,  Show that A² = I₃.

If [1 1 x] `[[1         0            2],[0           2         1],[2            1           0]] [[1],[1],[1]]` = 0, find x.

 If `[[2     3],[5      7]] [[1      -3],[-2       4]]-[[-4      6],[-9        x]]` find x.

If [x 4 1] `[[2       1          2],[1         0          2],[0       2 -4]]`  `[[x],[4],[-1]]` = 0, find x.

If [1 −1 x] `[[0       1           -1],[2           1             3],[1          1             1]]   [[0],[1],[1]]=`= 0, find x.

\[A = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix} and \text{ I }= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\],  then prove that A² − A + 2I = O.

\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

If A= `[[3      1],[-1        2]]` show that A² − 5A + 7I₂ = O

\[A = \begin{bmatrix}2 & 3 \\ - 1 & 0\end{bmatrix}\],show that A² − 2A + 3I₂ = O

Show that the matrix  \[A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}\]satisfies the equation A³ − 4A² + A = O

Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\]  is  root of the equation A² − 12A − I = O

If \[A = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}\] , find A² − 5A − 14I.

\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\]show that A² − 5A + 7I = O use this to find A⁴.

If A=`[[3,-2],[4,-2]] ` , find k such that A² = kA − 2I₂

If A=`[[1         0],[-1       7]]`, find k such that A² − 8A + kI = 0.

If\[A = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}\] f (x) = x² − 2x − 3, show that f (A) = 0

If A=`[[2,3],[1,2]]`  and I= `[[1,0],[0,1]]` then find λ, μ so that A² = λA + μI

Find the value of x for which the matrix product`[[2       0           7],[0          1            0],[1       -2       1]]` `[[-x         14x          7x],[0         1            0],[x           -4x             -2x]]`equal an identity matrix.

Solve the matrix equations:

`[x1][[1,0],[-2,-3]][[x],[5]]=0`

Solve the matrix equations:

`[1  2   1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`

Solve the matrix equations:

`[[],[x-5-1],[]][[1,0,2],[0,2,1],[2,0,3]] [[x],[4],[1]]=0`

Solve the matrix equations:

[2x 3] `[[1       2],[-3      0]] , [[x],[8]]=0`

If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A² − 4A + 3I₃.

If f (x) = x² − 2x, find f (A), where A= `[[0,1,2],[4,5,0],[0,2 ,3]]`

If f (x) = x³ + 4x² − x, find f (A), where\[A = \begin{bmatrix}0 & 1 & 2 \\ 2 & - 3 & 0 \\ 1 & - 1 & 0\end{bmatrix}\]

If A= `[[1           0           2],[0        2           1],[2            0           3]]` , then show that A is a root of the polynomial f (x) = x³ − 6x² + 7x + 2.

`A=[[1,2,2],[2,1,2],[2,2,1]]`, then prove that A² − 4A − 5I = 0

`A=[[3,2, 0],[1,4,0],[0,0,5]]` show that A² − 7A + 10I₃ = 0

Without using the concept of inverse of a matrix, find the matrix `[[x       y],[z       u]]` such that
`[[5     -7],[-2         3]][[x        y],[z         u]]=[[-16       -6],[7                   2]]`

Find the matrix A such that `[[1     1],[0       1]]A=[[3        3         5],[1       0          1]]`

Find the matrix A such that `A=[[1,2,3],[4,5,6]]=`  `[[-7,-8,-9],[2,4,6]]`

Find the matrix A such that `[[4],[1],[3]]  A=[[-4,8,4],[-1,2,1],[-3,6,3]]`

Find the matrix A such that    [2  1  3 ] `[[-1,0,-1],[-1,1,0],[0,1,1]] [[1],[0],[-1]]=A`

Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`

Find the matrix A such that `=[[1,2,3],[4,5,6]]=[[-7,-8,-9],[2,4,6],[11,10,9]]`

Find a 2 × 2 matrix A such that `A=[[1,-2],[1,4]]=6l_2`

If `A=[[0,0],[4,0]]` find `A^16`

If `A=[[0,-x],[x,0]],[[0,1],[1,0]]` and `x^2=-1,` then  show that `(A+B)^2=A^2+B^2`

`A=[[1,0,-3],[2,1,3],[0,1,1]]`then verify that A² + A = A(A + I), where I is the identity matrix.

`A=[[3,-5],[-4,2]]` then find A² − 5A − 14I. Hence, obtain A³

 If `P(x)=[[cos x,sin x],[-sin x,cos x]],` then show that `P(x),P(y)=P(x+y)=P(y)P(x).`

If `P=[[x,0,0],[0,y,0],[0,0,z]]` and `Q=[[a,0,0],[0,b,0],[0,0,c]]` prove that `PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP`

`A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A² − 5A + 4I and hence find a matrix X such that A² − 5A + 4I + X = 0.

If `A=[[1,1],[0,1]] ,` Prove that `A=[[1,n],[0,1]]` for all positive integers n.

If\[A = \begin{bmatrix}a & b \\ 0 & 1\end{bmatrix}\], prove that\[A^n = \begin{bmatrix}a^n & b( a^n - 1)/a - 1 \\ 0 & 1\end{bmatrix}\] for every positive integer n .

If `A=[[cos θ, i sinθ],[i sinθ,cosθ]]` then prove by principle of mathematical induction that `A^n=[[cos  nθ,i sinθ],[i sin nθ,cos nθ]]` for all `n  ∈ N.`

\[A = \begin{bmatrix}\cos \alpha + \sin \alpha & \sqrt{2}\sin \alpha \\ - \sqrt{2}\sin \alpha & \cos \alpha - \sin \alpha\end{bmatrix}\] ,prove that

\[A^n = \begin{bmatrix}\text{cos n α} + \text{sin n α}  & \sqrt{2}\text{sin n  α} \\ - \sqrt{2}\text{sin n α} & \text{cos n α} - \text{sin  n  α} \end{bmatrix}\] for all n ∈ N.

Let `A= [[1,1,1],[0,1,1],[0,0,1]]` Use the principle of mathematical introduction to show  that `A^n [[1,n,n(n+1)//2],[0,1,1],[0,0,1]]` for every position integer n.

Give examples of matrices
A and B such that AB ≠ BA

Give examples of matrices

 A and B such that AB = O but A ≠ 0, B ≠ 0.

Give examples of matrices

A and B such that AB = O but BA ≠ O.

Give examples of matrices

 A, B and C such that AB = AC but B ≠ C, A ≠ 0.

If A and B are square matrices of the same order, explain, why in general

(A + B)² ≠ A² + 2AB + B²

If A and B are square matrices of the same order, explain, why in general

(A − B)² ≠ A² − 2AB + B²

If A and B are square matrices of the same order, explain, why in general

 (A + B) (A − B) ≠ A² − B²

Let `A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]]` and `C=[[4,2],[-3,5],[5,0]]`Verify that AB = AC though B ≠ C, A ≠ O.

In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters. The cost per contact (in paise) is given matrix A as

      Cost per contact

`A=[[40],[100],[50]]` `[["Teliphone"] ,["House call "],[" letter"]]`

The number of contacts of each type made in two cities X and Y is given in matrix B as

       Telephone   House call    Letter

`B= [[    1000, 500,      5000],[3000,1000,     10000                ]]` 

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of
(i) Rs 1800 

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of(ii) Rs 2000

To promote making of toilets for women, an organisation tried to generate awarness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below:

(i) ₹50       (ii) ₹20       (iii) ₹40

The number of attempts made in three villages X, Y and Z are given below:

          (i)               (ii)              (iii)
X      400              300             100
Y      300              250               75
Z      500              400             150

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
\[A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array} \text{Telephone}\\{\text{House calls }}\\ \text{Letters}\end{array}\]

The number of contacts of each type made in two cities X and Y is given in the matrix B as

\[\begin{array}"Telephone & House calls & Letters\end{array}\]

\[B = \begin{bmatrix}1000 & 500 & 5000 \\ 3000 & 1000 & 10000\end{bmatrix}\begin{array} \\City   X \\ City Y\end{array}\]

Find the total amount spent by the party in the two cities.

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Let  `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that 

 (2A)^T = 2A^T

Let  `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that 

 (A + B)^T = A^T + B^T

Let  `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that 

(A − B)^T = A^T − B^T

Let  `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that 

If `A= [[3],[5],[2]]` And B=[1  0   4] , Verify that `(AB)^T=B^TA^T` 

Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that   (A + B)^T = A^T + B^T

A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} and B = \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix} . Find A^T, B^T and verify that , 

(A B)^T = B^T + A^T

Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that (2A)^T = 2A^T.

If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)^T = B^T A^T

If\[A = \begin{bmatrix}\cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha\end{bmatrix}\] , then verify that A^T A = I₂.

If l_i, m_i, n_i, i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^T = I, where \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]

If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − A^T is a skew-symmetric matrix.

Express the matrix \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\]  as the sum of a symmetric and a skew-symmetric matrix.

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result:

If  \[A = \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}\] write AA^T.

If \[A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\] , find x satisfying 0 < x < \[\frac{\pi}{2}\] when A + A^T = I

If  \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\]  , find AA^T

If \[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\] satisfies A⁴ = λA, then write the value of λ.

If  \[A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] , find A³.

If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A⁴.

If  `[x        2]  [[3],[4]] = 2` , find x

Write matrix A satisfying   ` A+[[2      3],[-1   4]] =[[3     6],[- 3     8]]`.

If   ` A =[a_ij]`    is a skew-symmetric matrix, then write the value of ` Σ_i  a_ij`

If A = [a_ij] is a skew-symmetric matrix, then write the value of  \[\sum_i \sum_j\]  a_ij.

Find the values of x and y, if \[2\begin{bmatrix}1 & 3 \\ 0 & x\end{bmatrix} + \begin{bmatrix}y & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}\]

If  \[\begin{bmatrix}x + 3 & 4 \\ y - 4 & x + y\end{bmatrix} = \begin{bmatrix}5 & 4 \\ 3 & 9\end{bmatrix}\] , find x and y

Find the value of x from the following: `[[2x - y          5],[ 3                         y ]]` = `[[6            5 ],[3         - 2\]]`

Find the value of y, if \[\begin{bmatrix}x - y & 2 \\ x & 5\end{bmatrix} = \begin{bmatrix}2 & 2 \\ 3 & 5\end{bmatrix}\]

Find the value of x, if  \[\begin{bmatrix}3x + y & - y \\ 2y - x & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ - 5 & 3\end{bmatrix}\]

if  \[\begin{bmatrix}2x + y & 3y \\ 0 & 4\end{bmatrix} = \begin{bmatrix}6 & 0 \\ 6 & 4\end{bmatrix}\]  , then find x.

If \[A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\] , find A + A^T.

If \[\begin{bmatrix}a + b & 2 \\ 5 & b\end{bmatrix} = \begin{bmatrix}6 & 5 \\ 2 & 2\end{bmatrix}\] , then find a.

If \[A = \begin{bmatrix}\cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\] is identity matrix, then write the value of α.

If  \[\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\begin{bmatrix}3 & 1 \\ 2 & 5\end{bmatrix} = \begin{bmatrix}7 & 11 \\ k & 23\end{bmatrix}\] ,then write the value of k.

If \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.

If \[\begin{bmatrix}x & x - y \\ 2x + y & 7\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 8 & 7\end{bmatrix}\]  , then find the value of y.

For a 2 × 2 matrix A = [a_ij] whose elements are given by 

If  \[x\binom{2}{3} + y\binom{ - 1}{1} = \binom{10}{5}\] , find the value of x.

If  \[\begin{bmatrix}9 & - 1 & 4 \\ - 2 & 1 & 3\end{bmatrix} = A + \begin{bmatrix}1 & 2 & - 1 \\ 0 & 4 & 9\end{bmatrix}\] , then find matrix A.

If  \[\begin{bmatrix}a - b & 2a + c \\ 2a - b & 3c + d\end{bmatrix} = \begin{bmatrix}- 1 & 5 \\ 0 & 13\end{bmatrix}\] , find the value of b.

For what value of x, is the matrix  \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\]  a skew-symmetric matrix?

If matrix  \[A = \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix}\]  and A² = pA, then write the value of p.

If  \[2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}\] , find x − y.

If \[\begin{bmatrix}x & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 2 & 0\end{bmatrix} = O\]  , find x.

If \[\begin{bmatrix}a + 4 & 3b \\ 8 & - 6\end{bmatrix} = \begin{bmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{bmatrix}\] , write the value of a − 2b.

If  \[\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix} = \begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}\] , write the value of (x + y + z).

Construct a 2 × 2 matrix A = [a_ij] whose elements a_ij are given by \[a_{ij} = \begin{cases}\frac{\left| - 3i + j \right|}{2} & , if i \neq j \\ \left( i + j \right)^2 & , if i = j\end{cases}\]

If  \[\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}\] , then write the value of (x, y).

Matrix A = \[\begin{bmatrix}0 & 2b & - 2 \\ 3 & 1 & 3 \\ 3a & 3 & - 1\end{bmatrix}\]  is given to be symmetric, find values of a and b.

If `[2     1       3]([-1,0,-1],[-1,1,0],[0,1,1])([1],[0],[-1])=A` , then write the order of matrix A.

`If A = ([3   5] , [7     9])` is written as A = P + Q, where as A = p + Q , Where  P is a symmetric matrix and Q is skew symmetric matrix , then wqrite the matrix P. 

Let A and B be matrices of orders 3 x 2 and 2 x 

If \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\] , then A² is equal to ___________ .

If `A=[[i,0],[0,i ]]` , n ∈ N, then A⁴ⁿ equals

If A and B are two matrices such that AB = A and BA = B, then B² is equal to

If AB = A and BA = B, where A and B are square matrices,  then

If A and B are two matrices such n  that AB = B and BA = A , `A^2 + B^2` is equal to

If  \[\begin{bmatrix}\cos\frac{2\pi}{7} & - \sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}\end{bmatrix}^k = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] then the least positive integral value of k is _____________.

If the matrix AB is zero, then

Let A = \[\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\], then Aⁿ is equal to

If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a 

If \[A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix} \text {and B} = \begin{bmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c^1 & c_2 & c_3\end{bmatrix}\]then AB is equal to

If  \[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\]then Aⁿ (where n ∈ N) equals 

If  \[A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] and AB = I₃, then x + y equals 

If \[A = \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}, B = \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\]and (A + B)² = A² + B²,   values of a and b are

If  \[A = \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}\]  is such that A² = I, then 

If S = [S_ij] is a scalar matrix such that s_ij = k and A is a square matrix of the same order, then AS = SA = ? 

If A is a square matrix such that A² = A, then (I + A)³ − 7A is equal to

If a matrix A is both symmetric and skew-symmetric, then

The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is

If A is a square matrix, then AA is a

If A and B are symmetric matrices, then ABA is

If \[A = \begin{bmatrix}5 & x \\ y & 0\end{bmatrix}\]  and A = A^T, then

If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined. Then, B is of the type 

If A = [a_ij] is a square matrix of even order such that a_ij = i² − j², then 

If \[A = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}\]  then A^T + A = I₂, if

If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\]  is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is  

Out of the given matrices, choose that matrix which is a scalar matrix: 

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is

Which of the given values of x and y make the following pairs of matrices equal? \[\begin{bmatrix}3x + 7 & 5 \\ y + 1 & 2 - 3x\end{bmatrix}, \begin{bmatrix}0 & y - 2 \\ 8 & 4\end{bmatrix}\] 

If \[A = \begin{bmatrix}0 & 2 \\ 3 & - 4\end{bmatrix}\]  and \[kA = \begin{bmatrix}0 & 3a \\ 2b & 24\end{bmatrix}\]  then the values of k, a, b, are respectively 

If \[I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\] then B equals ) 

The trace of the matrix \[A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}\], is

If A = [a_ij] is a scalar matrix of order n × n such that a_ii = k, for all i, then trace of A is equal to

The matrix  \[A = \begin{bmatrix}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{bmatrix}\] is a

The number of possible matrices of order 3 × 3 with each entry 2 or 0 is 

If \[\begin{bmatrix}2x + y & 4x \\ 5x - 7 & 4x\end{bmatrix} = \begin{bmatrix}7 & 7y - 13 \\ y & x + 6\end{bmatrix}\] 

If A is a square matrix such that A² = I, then (A − I)³ + (A + I)³ − 7A is equal to 

If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is 

If A is a matrix of order m × n and B is a matrix such that AB^T and B^TA are both defined, then the order of matrix B is 

If A and B are matrices of the same order, then AB^T − BA^T is a 

If matrix  \[A = \left[ a_{ij} \right]_{2 \times 2}\] where 

If \[A = \frac{1}{\pi}\begin{bmatrix}\sin^{- 1} \left( \ pix \right) & \ tan^{- 1} \left( \frac{x}{\pi} \right) \\ \sin^{- 1} \left( \frac{x}{\pi} \right) & \cot^{- 1} \left( \ pix \right)\end{bmatrix}, B = \frac{1}{\pi}\begin{bmatrix}- \cos^{- 1} \left( \ pix \right) & \tan^{- 1} \left( \frac{x}{\pi} \right) \\ \sin^{- 1} \left( \frac{x}{\pi} \right) & - \tan^{- 1} \left( \ pix \right)\end{bmatrix}\]

A − B is equal to

If A and B are square matrices of the same order, then (A + B)(A − B) is equal to 

If  \[A = \begin{bmatrix}2 & - 1 & 3 \\ - 4 & 5 & 1\end{bmatrix}\text{ and B }= \begin{bmatrix}2 & 3 \\ 4 & - 2 \\ 1 & 5\end{bmatrix}\] then

The matrix  \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a 

The matrix   \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is