Matrix operations — addition, subtraction (conditions): Exercises

This is a self-testing page for Matrix operations — addition, subtraction (conditions). Work each problem on paper first, then open its solution. The exercises climb from recognition (can you spot when addition is even legal?) up to mastery (can you chain every property together?).

Before we start, one reminder of the single rule that governs everything below:

Recall What must be true before you can add or subtract two matrices?

The two matrices must have the same order — the same number of rows and the same number of columns. Only then does every entry $(i, j)$ in one matrix has a partner $(i, j)$ in the other. See Matrix Notation and Terminology for what "order" means.

A quick picture of the whole idea — two grids can only combine if they line up cell-for-cell:

Figure — Matrix operations — addition, subtraction (conditions)

Notation used on this page

Everything here uses only symbols already built in the parent note. Just to be safe:

Level 1 — Recognition

Exercise 1.1 (L1)

For each pair, state whether $A + B$ is defined. If yes, give the order of the answer.

(a) $A$ is $2 \times 3$ , $B$ is $2 \times 3$ . (b) $A$ is $3 \times 2$ , $B$ is $2 \times 3$ . (c) $A$ is $4 \times 1$ , $B$ is $4 \times 1$ . (d) $A$ is $1 \times 5$ , $B$ is $5 \times 1$ .

Recall Solution 1.1

Addition is defined only when both orders are identical.

(a) Both $2 \times 3$ → ✓ defined, result is $2 \times 3$ .
(b) $3 \times 2$ vs $2 \times 3$ → ✗ undefined (rows and columns are swapped, not equal).
(c) Both $4 \times 1$ → ✓ defined, result is $4 \times 1$ .
(d) $1 \times 5$ vs $5 \times 1$ → ✗ undefined. A single row and a single column are not the same shape.

Exercise 1.2 (L1)

Given $A = [35 - 2 1], B = [- 3 - 5 2 - 1]$ Without heavy computation, what is $A + B$ ? What special matrix is it?

Recall Solution 1.2

Every entry of $B$ is the sign-flip of the matching entry of $A$ , so $B = - A$ . $A + B = [3 + (- 3) 5 + (- 5) - 2 + 2 1 + (- 1)] = [0000] = O$ This is the zero matrix, and $B$ is the additive inverse of $A$ .

Level 2 — Application

Exercise 2.1 (L2)

Compute $A + B$ : $A = [21 - 3 0 5 - 2], B = [4 - 3 17 - 1 6]$

Recall Solution 2.1

Same order $2 \times 3$ → defined. Add cell by cell: $A + B = [2 + 4 1 + (- 3) - 3 + 1 0 + 7 5 + (- 1) - 2 + 6] = [6 - 2 - 2 7 44]$

Exercise 2.2 (L2)

Compute $P - Q$ : $P = 530 - 1 24, Q = 2 - 1 5 31 - 2$

Recall Solution 2.2

Both $3 \times 2$ → defined. Subtraction is "add the sign-flip", so $d_{ij} = p_{ij} - q_{ij}$ : $P - Q = 5 - 2 3 - (- 1) 0 - 5 - 1 - 3 2 - 1 4 - (- 2) = 34 - 5 - 4 16$ Watch the double-negatives: $3 - (- 1) = 4$ and $4 - (- 2) = 6$ .

Exercise 2.3 (L2)

With $A, B$ from Exercise 2.1, compute $B - A$ . Compare it to $A - B$ .

Recall Solution 2.3

$A - B = [2 - 4 1 - (- 3) - 3 - 1 0 - 7 5 - (- 1) - 2 - 6] = [- 2 4 - 4 - 7 6 - 8]$ $B - A = [2 - 4 47 - 6 8]$ Every entry of $B - A$ is the negative of $A - B$ : this is anti-commutativity, $B - A = - (A - B)$ . Subtraction does not commute.

Level 3 — Analysis

Exercise 3.1 (L3)

Find scalars $x$ and $y$ so that $[x 3 2 y] + [4 - 3 - 2 5] = [7009]$

Recall Solution 3.1

Matrix equality means each cell must match. Read the two cells that contain unknowns:

Cell $(1, 1)$ : $x + 4 = 7 \Rightarrow x = 3$ .
Cell $(2, 2)$ : $y + 5 = 9 \Rightarrow y = 4$ .

(Check the other cells for consistency: $2 + (- 2) = 0$ ✓, $3 + (- 3) = 0$ ✓.) So $x = 3, y = 4$ . Notice this is really a tiny System of Linear Equations hidden inside one matrix equation.

Exercise 3.2 (L3)

Matrix $M$ is $2 \times 3$ and matrix $N$ is $3 \times 2$ . Is $M + N^{T}$ defined? Is it the same operation as "adding $M$ and $N$ "? Explain.

Recall Solution 3.2

Defined? $N^{T}$ turns $N$ from $3 \times 2$ into $2 \times 3$ . Now $M$ and $N^{T}$ are both $2 \times 3$ , so $M + N^{T}$ is defined.
Same as $M + N$ ? No. $M + N$ is undefined to begin with ( $2 \times 3$ vs $3 \times 2$ ). More importantly, transposing $N$ changes the meaning of each entry: entry $(1, 2)$ of $N$ ("row-item 1, col-item 2") lands at position $(2, 1)$ of $N^{T}$ . So $M + N^{T}$ is a legitimate but different computation — it is not "the sum of $M$ and $N$ ". See Transpose of a Matrix.

Exercise 3.3 (L3)

Let $A = [10 - 2 3]$ . Find the matrix $X$ satisfying $A + X = O$ .

Recall Solution 3.3

We want the additive inverse: $X = - A$ , obtained by flipping every sign. $X = [- 1 0 2 - 3]$ Check: $A + X = [1 - 1 0 + 0 - 2 + 2 3 - 3] = [0000] = O$ ✓. Note $- 0 = 0$ , so the zero entry stays put.

Level 4 — Synthesis

Exercise 4.1 (L4)

Given $A = [10 2 - 1], B = [32 - 1 4]$ compute $2 A + 3 B$ . (Uses Scalar Multiplication of Matrices combined with addition.)

Recall Solution 4.1

Scale first, then add cell by cell: $2 A = [20 4 - 2], 3 B = [96 - 3 12]$ $2 A + 3 B = [2 + 9 0 + 6 4 - 3 - 2 + 12] = [116110]$

Exercise 4.2 (L4)

Using $A, B$ from 4.1, verify the distributive law $k (A + B) = k A + k B$ for $k = 2$ .

Recall Solution 4.2

Left side: $A + B = [4213]$ , so $2 (A + B) = [8426]$ . Right side: $2 A = [20 4 - 2]$ , $2 B = [64 - 2 8]$ , so $2 A + 2 B = [8426]$ . Both sides equal $[8426]$ ✓. They match because for every entry $2 (a_{ij} + b_{ij}) = 2 a_{ij} + 2 b_{ij}$ — the distributive law of ordinary numbers, applied cell by cell.

Exercise 4.3 (L4)

Solve for matrix $X$ : $2 X - A = B$ , using $A, B$ from 4.1.

Recall Solution 4.3

Treat it like ordinary algebra, but every step is a matrix operation:

Add $A$ to both sides: $2 X = B + A$ .
$B + A = [4213]$ .
Multiply by scalar $\frac{1}{2}$ : $X = \frac{1}{2} [4213] = [21 \frac{1}{2} \frac{3}{2}]$ .

Check: $2 X - A = [4213] - [10 2 - 1] = [32 - 1 4] = B$ ✓.

Level 5 — Mastery

Exercise 5.1 (L5)

Three shops record weekly units of two products. Monday's grid and Tuesday's grid are $M = 1075438 (shops \times products), T = 694215$ On Wednesday every shop returns the amount stored in $R = 120013$ (a loss). Find the net two-day total $N = M + T - R$ , and confirm the operation is defined.

Recall Solution 5.1

All three matrices are $3 \times 2$ (same "shops × products" layout), so the whole expression is defined and stays $3 \times 2$ . Add $M + T$ first: $M + T = 161696413$ Subtract $R$ : $N = (M + T) - R = 16 - 1 16 - 2 9 - 0 6 - 0 4 - 1 13 - 3 = 151496310$ Every cell keeps its meaning "that shop, that product", which is exactly why element-wise combining is valid here.

Exercise 5.2 (L5)

Prove, entry by entry, that for any three $m \times n$ matrices, $(A + B) - C = A + (B - C)$ .

Recall Solution 5.2

All three share order $m \times n$ , so every operation below is defined and produces an $m \times n$ matrix; we only need to show the $(i, j)$ entries agree.

Left side entry: $((A + B) - C)_{ij} = (a_{ij} + b_{ij}) - c_{ij}$ .
Right side entry: $(A + (B - C))_{ij} = a_{ij} + (b_{ij} - c_{ij})$ .

For ordinary real numbers, addition/subtraction is associative: $(a + b) - c = a + (b - c)$ . Since this holds in every cell $(i, j)$ , the two matrices are equal. ∎ This is the deep reason all matrix-addition laws work: they are just the number laws applied simultaneously to every entry.

Exercise 5.3 (L5)

Find all $2 \times 2$ matrices $X$ with $X = X^{T}$ and $X + X = [46 6 - 2]$ .

Recall Solution 5.3

Write $X = [a c b d]$ .

$X + X = 2 X = [2 a 2 c 2 b 2 d] = [46 6 - 2]$ , giving $a = 2, b = 3, c = 3, d = - 1$ .
Symmetry condition $X = X^{T}$ needs $b = c$ ; here $b = c = 3$ ✓, so it is automatically satisfied.

Unique answer: $X = [23 3 - 1]$ .

Connections

Matrix operations — addition, subtraction (conditions) — parent rules these exercises drill.
Zero Matrix and Identity Matrix — the $O$ used in 1.2 and 3.3.
Scalar Multiplication of Matrices — powers L4 and 5.3.
Transpose of a Matrix — the trap in 3.2 and symmetry in 5.3.
System of Linear Equations — hidden inside the matrix equations of L3–L4.
Matrix Multiplication — the different operation students confuse with scaling.