Question:
If $A = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$ then $A^{16}$ is equal to :
If $A = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$ then $A^{16}$ is equal to :
Updated On: Jul 6, 2022
- $\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$
- $\begin{bmatrix}0& 1\\ 1&0\end{bmatrix}$
- $\begin{bmatrix} - 1 &0 \\ 0 & 1 \end{bmatrix}$
- $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
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The Correct Option is D
Solution and Explanation
We have $A = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix} $
Now, $A^{2} = A.A = \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix} $
$ = \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} = - I $
where $I = \begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ is identity matrix
$ \left(A^{2}\right)^{8} = \left(-I\right)^{8} = I$
Hence, $ A^{16} = I $
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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.