Question:
If $A = \begin{bmatrix}0&5\\ 0&0\end{bmatrix} $ and $f\left(X\right) =1+x+x^{2} +....+x^{16} $, then $f(A) = $
If $A = \begin{bmatrix}0&5\\ 0&0\end{bmatrix} $ and $f\left(X\right) =1+x+x^{2} +....+x^{16} $, then $f(A) = $
Updated On: Jul 6, 2022
- $O$
- $ \begin{bmatrix} 1 &5\\ 0& 1 \end{bmatrix} $
- $ \begin{bmatrix} 1 &5\\ 0&0\end{bmatrix} $
- $ \begin{bmatrix}0&5\\ 1 &1\end{bmatrix} $
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The Correct Option is B
Solution and Explanation
Clearly $f(A)=I+A+A^2+......+A^{16}$
$A^{2} = AA = \begin{bmatrix}0&5\\ 0&0\end{bmatrix} \begin{bmatrix}0&5\\ 0&0\end{bmatrix} = \begin{bmatrix}0&0\\ 0&0\end{bmatrix}= O $
$A^3=0,A^4=0,.......A^{16}=0$
$\therefore$$f(A)$$ = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} + \begin{bmatrix}0&5\\ 0&0\end{bmatrix}+0+0+.....+0$
$=\begin{bmatrix}1&5\\ 0&1\end{bmatrix}$
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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.