Q.Find matrix X such that AX=B, where $A=\begin{bmatrix}1&-2\\-2&1\end{bmatrix}$ and $B=\begin{bmatrix}-3\\-1\end{bmatrix}$.

Given $A=\begin{bmatrix}1&-2\\-2&1\end{bmatrix}$, $B=\begin{bmatrix}-3\\-1\end{bmatrix}$. Let $X=\begin{bmatrix}x_1\\x_2\end{bmatrix}$.

Expand AX=B:

$\begin{bmatrix}1&-2\\-2&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_1-2x_2\\-2x_1+x_2\end{bmatrix}=\begin{bmatrix}-3\\-1\end{bmatrix}$

This gives two simultaneous equations:

$x_1-2x_2=-3$ …(i)

$-2x_1+x_2=-1$ …(ii)

Solve: from (i), $x_1=2x_2-3$. Substitute into (ii):

$-2(2x_2-3)+x_2=-1 \Rightarrow -4x_2+6+x_2=-1 \Rightarrow -3x_2=-7 \Rightarrow x_2=\dfrac{7}{3}$ …

Write X as an unknown 2×1 column matrix, expand AX=B into two linear equations, and solve them simu …

• Trying to 'divide' B by A entrywise instead of solving the linear system AX=B (or using $A^{-1}$) properly.
• Sign errors in the elimination step when substituting $x_1=2x_2-3$ into the second equation. …