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  4. If [3&1 4&1] X = [5&-1 2&3] then X is equal to:

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Q. If $\begin{bmatrix}3&1\\ 4&1\end{bmatrix} X = \begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $ then X is equal to:

Matrices

Solution:

Given : $\begin{bmatrix}3&1\\ 4&1\end{bmatrix} X = \begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $
Thus $X = \begin{bmatrix}3&1\\ 4&1\end{bmatrix}^{-1} \begin{bmatrix}5&-1\\ 2&3\end{bmatrix}$
Let $ A = \begin{bmatrix}3&1\\ 4&1\end{bmatrix} $
$a_{11}$ = co-factor of $a_{11} = 1$
$a_{12}$ = co-factor of $a_{12} = (-1)^{1+2}. 4 = - 4$
$a_{21}$ = co-factor of $a_{21} = (-1)^{2+1} . 1 = - 1$
$a_{22}$ = co-factor of $a_{22} = 3 $
$| A | = 3 - 4 = -1 $
So $ A^{-1} = \frac{\begin{bmatrix}1&-4\\ -1&3\end{bmatrix}}{-1} = \begin{bmatrix}-1&1\\ 4&-3\end{bmatrix}$
Thus $ X = \begin{bmatrix}-1&1\\ 4&-13\end{bmatrix}\begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $
$ X = \begin{bmatrix}-3&4\\ 14&-13\end{bmatrix} $