Question:
if $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix} , B =\begin{bmatrix}x&1\\ y&-1\end{bmatrix}$ and $\left(A + B\right)^{2} =A^{2} + B^{2},$ then $x + y$ =
if $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix} , B =\begin{bmatrix}x&1\\ y&-1\end{bmatrix}$ and $\left(A + B\right)^{2} =A^{2} + B^{2},$ then $x + y$ =
Updated On: Jul 6, 2022
- $2$
- $3$
- $4$
- $5$
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The Correct Option is D
Solution and Explanation
$ \left(A + B\right)^{2 }= A^{2 }+ B^{2} \Rightarrow AB+ BA= 0$
$\Rightarrow\begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}\begin{bmatrix}x&1\\ y&-1\end{bmatrix}+\begin{bmatrix}x&1\\ y&-1\end{bmatrix}\begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}=0$
$\Rightarrow\begin{bmatrix}x-y&2\\ 2x-y&3\end{bmatrix}+\begin{bmatrix}x+2&-x-1\\ y-2&-y+1\end{bmatrix}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}$
$\Rightarrow2x-y+2=0....\left(i\right), -x+1=0....\left(ii\right)$
$2x-2=0....\left(iii\right) and-y+4 =0\backslash,\backslash,....\left(iv\right)$
From $\left(ii\right), x = 1$ and from $\left(iv\right), y = 4$
Now, $x + y = 1 + 4 = 5 $
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.