Q.If $X = \begin{bmatrix} 3 & 1 & -1 \\ 5 & -2 & -3 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 & 1 & -1 \\ 7 & 2 & 4 \end{bmatrix}$, find

Concept: Matrix Addition — matrices of the same order are added (or subtracted) by adding (or subtracting) corresponding entries.

Step 1 — $X+Y$:

Add entrywise:

$X+Y = \begin{bmatrix} 3+2 & 1+1 & -1+(-1) \\ 5+7 & -2+2 & -3+4 \end{bmatrix} = \begin{bmatrix} 5 & 2 & -2 \\ 12 & 0 & 1 \end{bmatrix}$.

Step 2 — $2X-3Y$:

First multiply each matrix by the scalar, then subtract:

$2X = \begin{bmatrix} 6 & 2 & -2 \\ 10 & -4 & -6 \end{bmatrix}$,

$3Y = \begin{bmatrix} 6 & 3 & -3 \\ 21 & 6 & 12 \end{bmatrix}$.

So $2X-3Y = \begin{bmatrix} 6-6 & 2-3 & -2-(-3) \\ 10-21 & -4-6 & -6-12 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 1 \\ -11 & -10 & -18 \end{bmatrix}$.

Step 3 — Matrix $Z$ such that $X+Y+Z=0$:

From $X+Y+Z=0$, we get $Z = -(X+Y)$. Using the result from Step 1: …

Concept First: Why Matrix Addition Works the Way It Does

Matrices are just rectangular arrays of numbers. When we add two matrices, we're combining corresponding positions — like adding two tables of data where each cell lines up with its twin. This only makes sense if both matrices have the same shape (same number of rows and columns). Here, both $X$ and $Y$ are $2 \times 3$ matrices, so we're good.

The same logic extends to scaling (multiplying every entry by a number) and to finding an unknown matrix that satisfies an equation — treat the matrix as a variable and solve entry by entry.

Step-by-Step Solution

1. Compute $X + Y$

Add each entry in $X$ to the entry in the same position in $Y$.

2. Compute $2X - 3Y$

First scale each matrix: multiply every entry of $X$ by 2, and every entry of $Y$ by 3. Then subtract corresponding entries.

Now subtract:

Method: Scalar multiples and combinations of same-order matrices

Use this for expressions like $X+Y$, $2X-3Y$, or solving $X+Y+Z=O$.

Steps

Step 1: Confirm equal order.

Every matrix in the combination must share one order; otherwise the operation is undefined.

Step 2: Apply each scalar to every entry.

$kX$ multiplies all entries of $X$ by $k$ — not just the diagonal.

Step 3: Add or subtract entrywise. …

Common Mistakes

Mistake 1: Sign slips when subtracting $3Y$.

Why it's wrong: for the $(1,3)$ entry, $-2-(-3)=+1$, not $-5$. Correct approach: rewrite subtraction of a negative as addition before combining.

Mistake 2: Scaling only some entries.

Why it's wrong: $2X$ multiplies every entry of $X$ by 2, not just the diagonal or first row. Correct approach: apply the scalar to all entries. …