Question:
Let $A=\begin{pmatrix}
1&-1 &1 \\[0.3em]
2 &1 &-3 \\[0.3em]
1 &1&1 \end{pmatrix}$ and $\,10\,B=\begin{pmatrix}
4&2 &2 \\[0.3em]
-5 &0 & \alpha \\[0.3em]
1 &-2&3 \end{pmatrix} $. If B is the inverse of A , then $\alpha$ is
Let $A=\begin{pmatrix}
1&-1 &1 \\[0.3em]
2 &1 &-3 \\[0.3em]
1 &1&1 \end{pmatrix}$ and $\,10\,B=\begin{pmatrix}
4&2 &2 \\[0.3em]
-5 &0 & \alpha \\[0.3em]
1 &-2&3 \end{pmatrix} $. If B is the inverse of A , then $\alpha$ is
Updated On: Jul 5, 2022
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The Correct Option is D
Solution and Explanation
Since B = A$^{-1}$ $\therefore$ BA = A$^{-1}$ A = I
$\therefore$ (10 B) A = 10 I = $\begin{bmatrix}10&0&0\\ 0&10&0\\ 0&0&10\end{bmatrix}$
Also (10 B) A$ = \begin{bmatrix}
4&2 &2 \\[0.3em]
-5 &0 & \alpha \\[0.3em]
1 &-2&3 \end{bmatrix} \begin{bmatrix}
1&-1 &1 \\[0.3em]
2 &1 &-3 \\[0.3em]
1 &1&1 \end{bmatrix}$
= $\begin{bmatrix}10&0&0\\ -5 + \alpha &5 + \alpha&-5 + \alpha\\ 0&0&10\end{bmatrix} = $ 101 if $\alpha$ = 5
Hence $\alpha$ = 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.
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