Question:
If $3A + 4B' = $$\begin{bmatrix}7&-10&17\\ 0&6&31\end{bmatrix} $ and $2B - 3A'$ $ \begin{bmatrix}-1&18\\ 4&0\\ -5&-7\end{bmatrix}$ then $B =$
If $3A + 4B' = $$\begin{bmatrix}7&-10&17\\ 0&6&31\end{bmatrix} $ and $2B - 3A'$ $ \begin{bmatrix}-1&18\\ 4&0\\ -5&-7\end{bmatrix}$ then $B =$
Updated On: Apr 18, 2024
- $\begin{bmatrix}-1&-18\\ 4&-16\\ -5&-7\end{bmatrix} $
- $\begin{bmatrix}1&3\\ -1&1\\ 2&4\end{bmatrix} $
- $\begin{bmatrix}1&3\\ -1&1\\ 2&-4\end{bmatrix} $
- $ \begin{bmatrix}1&-3\\ -1&1\\ 2&4\end{bmatrix} $
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The Correct Option is B
Solution and Explanation
$3A+4B' =\left[\begin{matrix}7&-10&17\\ 0&6&31\end{matrix}\right] \quad \dots\left(1\right)$
$\left(2B-3A'\right)'=\left(2B\right)'-\left(3A'\right)'=2B'-3A$
$\Rightarrow\, 2B' -3A=\left[\begin{matrix}-1&4&-5\\ 18&0&-7\end{matrix}\right]\quad\dots\left(2\right)$
Adding 1 and 2, we get $6B'=\left[\begin{matrix}6&-6&12\\ 18&6&24\end{matrix}\right]$
$\Rightarrow\, B' =\left[\begin{matrix}1&-1&2\\ 3&1&4\end{matrix}\right] \, \therefore\, B=\left[\begin{matrix}1&3\\ -1&1\\ 2&4\end{matrix}\right]$
$\left(2B-3A'\right)'=\left(2B\right)'-\left(3A'\right)'=2B'-3A$
$\Rightarrow\, 2B' -3A=\left[\begin{matrix}-1&4&-5\\ 18&0&-7\end{matrix}\right]\quad\dots\left(2\right)$
Adding 1 and 2, we get $6B'=\left[\begin{matrix}6&-6&12\\ 18&6&24\end{matrix}\right]$
$\Rightarrow\, B' =\left[\begin{matrix}1&-1&2\\ 3&1&4\end{matrix}\right] \, \therefore\, B=\left[\begin{matrix}1&3\\ -1&1\\ 2&4\end{matrix}\right]$
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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.