Q.If 2X + 3Y = (2 3; 4 0) and 3X + 2Y = (2 -2; -1 5), find X and Y. OR If A = (3 1; -1 2), show that A² - 5A + 7I = 0.

Two matrix equations in $X,Y$ are solved exactly like two scalar simultaneous equations — eliminate one unknown by scaling and subtracting.

Step 1. Given $2X+3Y=\begin{bmatrix}2&3\\4&0\end{bmatrix}$ (i) and $3X+2Y=\begin{bmatrix}2&-2\\-1&5\end{bmatrix}$ (ii).

Step 2. Eliminate $X$: multiply (i) by $3$ and (ii) by $2$:

$6X+9Y=\begin{bmatrix}6&9\\12&0\end{bmatrix},\qquad 6X+4Y=\begin{bmatrix}4&-4\\-2&10\end{bmatrix}.$

Subtract: $5Y=\begin{bmatrix}2&13\\14&-10\end{bmatrix}$, so $Y=\begin{bmatrix}2/5 & 13/5\\ 14/5 & -2\end{bmatrix}$.

Step 3. Substitute back into (i): $2X = \begin{bmatrix}2&3\\4&0\end{bmatrix}-3Y=\begin{bmatrix}2&3\\4&0\end{bmatrix}-\begin{bmatrix}6/5&39/5\\42/5&-6\end{bmatrix}=\begin{bmatrix}4/5&-24/5\\-22/5&6\end{bmatrix}$. …