Question:
If $A = \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix} $ then $\left(A^{2} - 5A\right)A^{-1} = $
If $A = \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix} $ then $\left(A^{2} - 5A\right)A^{-1} = $
Updated On: Jul 9, 2024
- $\begin{bmatrix}4&2&3\\ -1&4&2\\ 1&2&1\end{bmatrix}$
- $\begin{bmatrix} - 4&2&3\\ -1& -4&2\\ 1&2& -1\end{bmatrix}$
- $\begin{bmatrix}-4&-1&1\\ 2&-4&2\\ 3&2&-1\end{bmatrix}$
- $\begin{bmatrix}-1&-2&1\\ 4&-2&-3\\ 1&4&-2\end{bmatrix}$
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The Correct Option is B
Solution and Explanation
we have
$ A = \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix}$
Now,$ \left(A^{2} -5A\right)A^{-1}$
$ = A^{2} \cdot A^{-1} - 5A \cdot A^{-1} = A - 5I $
$= \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix}-\begin{bmatrix}5&0&0\\ 0&5&0\\ 0&0&5\end{bmatrix} $
$= \begin{bmatrix}-4&2&3\\ -1&-4&2\\ 1&2&-1\end{bmatrix}$
$ A = \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix}$
Now,$ \left(A^{2} -5A\right)A^{-1}$
$ = A^{2} \cdot A^{-1} - 5A \cdot A^{-1} = A - 5I $
$= \begin{bmatrix}1&2&3\\ -1&1&2\\ 1&2&4\end{bmatrix}-\begin{bmatrix}5&0&0\\ 0&5&0\\ 0&0&5\end{bmatrix} $
$= \begin{bmatrix}-4&2&3\\ -1&-4&2\\ 1&2&-1\end{bmatrix}$
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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.