Q.If $X$ and $Y$ are $2 \times 2$ matrices, then solve the following matrix equations for $X$ and $Y$: $2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix}$, $3X + 2Y = \begin{bmatrix} -2 & 2 \\ 1 & -5 \end{bmatrix}$.

Concept: Matrix Equation Solving — treat the system as two linear equations in the matrix unknowns $X$ and $Y$, and eliminate one variable.

Step 1: Multiply the first equation by 2 and the second by 3 to align coefficients of $Y$:

Step 2: Subtract the first from the second to eliminate $Y$:

Step 3: Divide by 5 to get $X$, then substitute back into $2X + 3Y = \text{(first matrix)}$ to find $Y$: …

The key insight is that matrix equations of this form behave exactly like ordinary linear equations — as long as we remember that matrix multiplication is not involved here. Each equation is a linear combination of the unknown matrices $X$ and $Y$ with scalar coefficients (2, 3, etc.). So we can solve them using the same elimination or substitution method we use for numbers.

Think of it this way: if we had $2x + 3y = a$ and $3x + 2y = b$, we’d solve for $x$ and $y$ by eliminating one variable. The same logic applies here — except $x$ and $y$ are $2 \times 2$ matrices, and $a$ and $b$ are the given $2 \times 2$ matrices on the right-hand sides.

Let’s label the equations for clarity:

We want to isolate $X$ and $Y$.

Step-by-step solution

Step 1: Eliminate $Y$ to find $X$.

Multiply equation (1) by 2 and equation (2) by 3, so that the coefficients of $Y$ become 6 and 6 — then subtract.

From (1):

From (2):

Now subtract the first result from the second:

The $6Y$ terms cancel, leaving:

Step 2: Compute $3B - 2A$.

First, $3B$:

Next, $2A$:

Now subtract:

So we have:

Step 3: Solve for $X$.

Divide both sides by 5 (i.e., multiply by $\frac{1}{5}$):

Step 4: Eliminate $X$ to find $Y$.

We can use a similar trick. Multiply equation (1) by 3 and equation (2) by 2, so the coefficients of $X$ become 6 and 6.

From (1):

From (2):

Now subtract the second from the first:

The $6X$ terms cancel, giving:

Step 5: Compute $3A - 2B$.

First, $3A$:

Next, $2B$: …

Method: Solving two simultaneous matrix equations

Use this when two equations are linear in unknown matrices $X$ and $Y$ with scalar coefficients.

Steps

Step 1: Treat the matrices like ordinary variables.

Because the coefficients are scalars, the same elimination you use for numbers works — no matrix inversion is needed.

Step 2: Scale to align a coefficient, then eliminate. …

Common Mistakes

Mistake 1: Reaching for matrix inversion or "division".

Why it's wrong: the coefficients of $X$ and $Y$ are scalars, so ordinary elimination works — no $A^{-1}$ is needed, and matrices cannot be "divided". Correct approach: scale and subtract the equations as with numbers.

Mistake 2: Sign errors in entrywise subtraction.

Why it's wrong: e.g. in $3B-2A$ the $(1,1)$ entry is $-6-4=-10$; mishandling the negatives corrupts $X$. Correct approach: subtract corresponding entries carefully. …