Q.If $A = \begin{bmatrix} 1 & -2 \\ 3 & -5 \\ -6 & 0 \end{bmatrix}$, $B = \begin{bmatrix} -1 & -2 \\ 4 & 2 \\ 1 & 5 \end{bmatrix}$ and $C = \begin{bmatrix} 2 & 4 \\ -1 & -4 \\ -3 & 6 \end{bmatrix}$, find the matrix $X$ such that $3A - 4B + 5X = C$.

Given $A = \begin{bmatrix} 1 & -2 \\ 3 & -5 \\ -6 & 0 \end{bmatrix}$, $B = \begin{bmatrix} -1 & -2 \\ 4 & 2 \\ 1 & 5 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 4 \\ -1 & -4 \\ -3 & 6 \end{bmatrix}$, and $3A-4B+5X=C$, so $5X = C-3A+4B$.

$3A = \begin{bmatrix} 3 & -6 \\ 9 & -15 \\ -18 & 0 \end{bmatrix}$, $\quad 4B = \begin{bmatrix} -4 & -8 \\ 16 & 8 \\ 4 & 20 \end{bmatrix}$

$C - 3A = \begin{bmatrix} 2-3 & 4-(-6) \\ -1-9 & -4-(-15) \\ -3-(-18) & 6-0 \end{bmatrix} = \begin{bmatrix} -1 & 10 \\ -10 & 11 \\ 15 & 6 \end{bmatrix}$ …

Isolate X as X = (C - 3A + 4B)/5, compute 3A and 4B, combine with C, …

• Isolating X incorrectly, e.g. writing $5X = 3A-4B-C$ instead of $C-3A+4B$. …