Q.Find $X$ and $Y$, if $X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$.

Concept: Matrix Addition — adding two matrices of the same order means adding their corresponding entries.

We are given:

Step 1: Add the two equations to eliminate $Y$:

Step 2: Divide by 2 to get $X$:

Step 3: Subtract the second equation from the first to eliminate $X$:

Step 4: Divide by 2 to get $Y$:

The problem gives you two matrix equations:

You need to find the matrices $X$ and $Y$. This looks exactly like a system of linear equations — except the “unknowns” are matrices, not numbers. But the same logic works: if you had $x+y = a$ and $x-y = b$, you’d add to get $2x = a+b$, then subtract to get $2y = a-b$. Matrix addition works entry-wise, so the same trick applies perfectly.

Let’s do it step by step.

1. Add the two equations Adding the left sides: $(X+Y) + (X-Y) = 2X$. Adding the right sides:

So $2X = \begin{bmatrix} 8 & 8 \\ 0 & 8 \end{bmatrix}$.

2. Solve for $X$ Divide every entry by 2:

3. Subtract the second equation from the first Left sides: $(X+Y) - (X-Y) = 2Y$. Right sides:

So $2Y = \begin{bmatrix} 2 & -4 \\ 0 & 10 \end{bmatrix}$.

4. Solve for $Y$ Divide by 2:

Method: Solving for two matrices from their sum and difference

Use this when given $X + Y$ and $X - Y$ and asked to find $X$ and $Y$.

Steps

Step 1: Add the two equations to eliminate $Y$.

$(X+Y) + (X-Y) = 2X$; adding the right-hand matrices entry-wise gives $2X$.

Step 2: Subtract to eliminate $X$.

$(X+Y) - (X-Y) = 2Y$; subtracting the right-hand matrices gives $2Y$.

Step 3: Halve each result.

$X = \tfrac12\,(\text{sum})$ and $Y = \tfrac12\,(\text{difference})$; optionally check by recomputing $X+Y$ and $X-Y$.

Common Mistakes

Mistake 1: Forgetting to divide by 2.

Why it's wrong: adding gives $2X$, not $X$; the final matrices must be halved. Correct approach: divide every entry of the sum and difference by $2$.

Mistake 2: Sign errors in the subtraction step.

Why it's wrong: $9 - (-1) = 10$, not $8$. Correct approach: subtract the second matrix entry-wise, minding negatives.

Mistake 3: Not verifying the answer.

Why it's wrong: a small slip is easy to miss without a check. Correct approach: confirm $X+Y$ and $X-Y$ reproduce the given matrices.