Advertisements
Advertisements
प्रश्न
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
विकल्प
a = 4, b = 1
a = 1, b = 4
a = 0, b = 4
a = 2, b = 4
उत्तर
a = 1, b = 4
\[Here, \]
\[ \left( A + B \right)^2 = A^2 + B^2 \]
\[ \Rightarrow A^2 + AB + BA + B^2 = A^2 + B^2 \]
\[ \Rightarrow AB + BA = O\]
\[ \Rightarrow AB = - BA\]
\[ \Rightarrow \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix} = - \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = - \begin{bmatrix}a + 2 & - a - 1 \\ b - 2 & - b + 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = \begin{bmatrix}- a - 2 & a + 1 \\ b + 2 & b - 1\end{bmatrix}\]
The corresponding elements of two equal matrices are equal .
\[ \Rightarrow a + 1 = \text{2 and b - 1} = 3\]
\[ \therefore a = \text{1 and b} = 4\]
APPEARS IN
संबंधित प्रश्न
Compute the indicated products:
`[[1 -2],[2 3]][[1 2 3],[-3 2 -1]]`
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]]`
Evaluate the following:
`([[1 3],[-1 -4]]+[[3 -2],[-1 1]])[[1 3 5],[2 4 6]]`
If A = `[[ab,b^2],[-a^2,-ab]]` , show that A2 = O
Let A =`[[-1 1 -1],[3 -3 3],[5 5 5]]`and B =`[[0 4 3],[1 -3 -3],[-1 4 4]]`
, compute A2 − B2.
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 [x 4 1] `[[2 1 2],[1 0 2],[0 2 -4]]` `[[x],[4],[-1]]` = 0, find x.
If [1 1 x] `[[1 0 2],[0 2 1],[2 1 0]] [[1],[1],[1]]` = 0, find x.
Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`
If `A=[[0,0],[4,0]]` find `A^16`
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.
The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.
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
Find the total cost incurred by the organisation for three villages separately, using matrices.
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 = 2AT
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 = 2AT.
If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)T = BT AT
If \[A = \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}\] write AAT.
If \[A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] , find A3.
If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
Construct a 2 × 2 matrix A = [aij] whose elements aij 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 A and B are two matrices such that AB = A and BA = B, then B2 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 A is a square matrix such that A2 = A, then (I + A)3 − 7A is equal to
If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
(a) nk (b) n + k (c) \[\frac{n}{k}\] (d) none of these
If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
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 matrix of order m × n and B is a matrix such that ABT and BTA are both defined, then the order of matrix B is
Disclaimer: option (a) and (d) both are the same.
Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2? Give reasons.
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn
If matrix A = [aij]2×2, where aij `{:(= 1 "if i" ≠ "j"),(= 0 "if i" = "j"):}` then A2 is equal to ______.
A square matrix where every element is unity is called an identity matrix.
Three schools DPS, CVC, and KVS decided to organize a fair for collecting money for helping the flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of Rs. 25, Rs.100, and Rs. 50 each respectively. The numbers of articles sold are given as
| School/Article | DPS | CVC | KVS |
| Handmade/fans | 40 | 25 | 35 |
| Mats | 50 | 40 | 50 |
| Plates | 20 | 30 | 40 |
Based on the information given above, answer the following questions:
- What is the total amount of money (in Rs.) collected by schools CVC and KVS?
