Question:
if $A \begin{bmatrix}1&2\\ 2&1\end{bmatrix}$and $f(x) = (1+x) (1 - x)$, then $f(A)$ is
if $A \begin{bmatrix}1&2\\ 2&1\end{bmatrix}$and $f(x) = (1+x) (1 - x)$, then $f(A)$ is
Updated On: Jul 6, 2022
- $-4 \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$
- $-8 \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$
- $4 \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$
- None of these
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The Correct Option is A
Solution and Explanation
Given, $f[x) = (1 + x) (1 - x) = 1 - x^2$
$\Rightarrow f(A) = I - A^2 (\because$ Put $x = A$)
$\Rightarrow f\left(A\right) = \begin{bmatrix}1&0\\ 0&1\end{bmatrix}-\left\{\begin{bmatrix}1&2\\ 2&1\end{bmatrix}\begin{bmatrix}1&2\\ 2&1\end{bmatrix}\right\}$
$=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}-\begin{bmatrix}5&4\\ 4&5\end{bmatrix}=\begin{bmatrix}-4&-4\\ -4&-4\end{bmatrix}$
$ = -4 \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$
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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.