Question

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

• A. $\begin{bmatrix}-3&4\\ 14&-13\end{bmatrix}$
• B. $ \begin{bmatrix}3&-4\\ -14&13\end{bmatrix}$
• C. $\begin{bmatrix}3&4\\ 14&13\end{bmatrix}$
• D. $\begin{bmatrix}-3&4\\ -14&13\end{bmatrix}$

Answer 1

The Correct Option is A

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} $