Advertisements
Advertisements
प्रश्न
उत्तर
\[Given: \hspace{0.167em} A = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}\]
\[ A^T = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\]
\[B = \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}\]
\[ B^T = \begin{bmatrix}1 & 2 \\ 4 & 5\end{bmatrix}\]
\[Now, \]
\[AB = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}1 + 6 & 4 + 15 \\ 2 + 8 & 8 + 20\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}7 & 19 \\ 10 & 28\end{bmatrix}\]
\[ \Rightarrow \left( AB \right)^T = \begin{bmatrix}7 & 10 \\ 19 & 28\end{bmatrix} . . . \left( 1 \right)\]
\[Also, \]
\[ B^T A^T = \begin{bmatrix}1 & 2 \\ 4 & 5\end{bmatrix}\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\]
\[ \Rightarrow B^T A^T = \begin{bmatrix}1 + 6 & 2 + 8 \\ 4 + 15 & 8 + 20\end{bmatrix}\]
\[ \Rightarrow B^T A^T = \begin{bmatrix}7 & 10 \\ 19 & 28\end{bmatrix} . . . \left( 2 \right)\]
\[ \therefore \left( AB \right)^T = B^T A^T \left[ \text{From eqs} . (1) and (2) \right]\]
APPEARS IN
संबंधित प्रश्न
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]]`
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
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 = O3×3.
If A =`[[2 -3 -5],[-1 4 5],[1 -3 -4]]` and B =`[[2 -2 -4],[-1 3 4],[1 2 -3]]`
, show that AB = A and BA = B.
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.
\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\] , Show that A2 = A.
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
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.
Solve the matrix equations:
`[1 2 1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`
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.
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 .
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) (A − B) ≠ A2 − B2
Three shopkeepers A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs Rs. 1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual's bill.
A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹ 2800 as interest. However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interes. Using matrix method, find the amount invested by the trust.
If \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
If A = [aij] is a 2 × 2 matrix such that aij = i + 2j, write A.
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 I is the identity matrix and A is a square matrix such that A2 = A, then what is the value of (I + A)2 = 3A?
For a 2 × 2 matrix A = [aij] whose elements are given by
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 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
If A = `[[3,9,0] ,[1,8,-2], [7,5,4]]` and B =`[[4,0,2],[7,1,4],[2,2,6]]` , then find the matrix `B'A'` .
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find X + Y
If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.
If A and B are square matrices of the same order, then [k (A – B)]′ = ______.
