Adding and Subtracting Matrices

Adding and subtracting matrices is the most direct thing you can do with them. There is no new arithmetic to learn — you already know how to add and subtract numbers. What you do need is one condition, checked before you start, and one habit: always work with corresponding entries, the ones sitting in the same row and the same column.

Concept The condition: equal order

Two matrices can be added or subtracted only when they have the same number of rows and the same number of columns — that is, when their orders are equal.

Both parts of the order have to agree. Matching on rows alone is not enough, and neither is holding the same total number of entries.

3 × 4 with 3 × 4 — allowed, the orders are identical.

2 × 2 with 2 × 2 — allowed.

3 × 4 with 4 × 3 — not allowed, even though both hold twelve entries.

3 × 4 with 3 × 2 — not allowed; the rows agree but the columns do not.

When the orders differ the operation is undefined. There is no partial answer and no convention for filling the gaps — some entry would be left without a partner, and the sum simply does not exist.

Concept Corresponding entries

Once the orders match, every entry in the first matrix has exactly one partner in the second: the entry in the same position. These are the corresponding entries, and they are the only pairs you ever combine.

The two shaded cells are corresponding entries. Combine those two numbers and the result goes into that same position in the answer.

The answer always has the same order as the two matrices you started with. Nothing changes shape.

Concept Adding matrices

Add the corresponding entries, one pair at a time: the first entry plus the entry facing it, the second plus the one facing it, and so on to the last.

\begin{bmatrix} 3 & -1 & 5 \\ 0 & 2 & 4 \end{bmatrix} + \begin{bmatrix} -2 & 7 & 1 \\ 6 & -3 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 6 & 6 \\ 6 & -1 & 4 \end{bmatrix}

Both matrices are 2 × 3, so the sum is defined and is itself 2 × 3. Six pairs of entries give six results.

Concept Subtracting matrices

Subtraction works exactly the same way and needs exactly the same condition. Take each entry of the first matrix and subtract the entry facing it in the second.

\begin{bmatrix} 3 & -1 & 5 \\ 0 & 2 & 4 \end{bmatrix} - \begin{bmatrix} -2 & 7 & 1 \\ 6 & -3 & 0 \end{bmatrix} = \begin{bmatrix} 5 & -8 & 4 \\ -6 & 5 & 4 \end{bmatrix}

Here the direction matters. Subtraction is not commutative: $A - B$ and $B - A$ give different matrices, each the negative of the other. Addition has no such issue — $A + B$ and $B + A$ agree.

Concept The general rule

(A \pm B)_{ij} = A_{ij} \pm B_{ij}

Read the subscripts as an address: $i$ is the row, $j$ is the column. The statement says that the entry at row $i$ , column $j$ of the answer is found from the entries at row $i$ , column $j$ of the two matrices — and from nothing else. Every position is handled independently, which is why the order of the result never changes.

Example A subtraction, entry by entry

Find $A - B$ , where

A = \begin{bmatrix} 3 & -1 & 5 \\ 0 & 2 & 4 \end{bmatrix}, \qquad B = \begin{bmatrix} -2 & 7 & 1 \\ 6 & -3 & 0 \end{bmatrix}

Both are 2 × 3, so the difference is defined and will also be 2 × 3.

Top row:

3 - (-2) = 5

-1 - 7 = -8

5 - 1 = 4

Bottom row:

0 - 6 = -6

2 - (-3) = 5

4 - 0 = 4

⟹

A - B = \begin{bmatrix} 5 & -8 & 4 \\ -6 & 5 & 4 \end{bmatrix}

Two of those steps subtracted a negative number and so increased the value. Subtracting −2 added 2; subtracting −3 added 3. That is where most errors in matrix subtraction come from.

Example Checking the order first

Can a 3 × 4 matrix be subtracted from a 4 × 3 matrix?

The first has 3 rows and 4 columns; the second has 4 rows and 3 columns.

Compare rows: 3 against 4 — not equal.

Compare columns: 4 against 3 — not equal either.

Both matrices hold twelve entries, but they are arranged differently.

⟹ the difference is undefined

Checking the order takes a moment and saves you from computing an answer that does not exist. The same check settles 3 × 4 with 3 × 2: the rows match, the columns do not, so that pair fails too.

Example Order of subtraction matters

Using the same $A$ and $B$ , find $B - A$ and compare it with $A - B$ .

Top row:

-2 - 3 = -5

7 - (-1) = 8

1 - 5 = -4

Bottom row:

6 - 0 = 6

-3 - 2 = -5

0 - 4 = -4

⟹

B - A = \begin{bmatrix} -5 & 8 & -4 \\ 6 & -5 & -4 \end{bmatrix}

Every entry is the exact opposite of the matching entry in $A - B$ . So $B - A$ is the negative of $A - B$ , which confirms that the order you write the two matrices in changes the answer.

Note Mistakes to avoid

Starting the arithmetic without checking that the orders are equal.

Assuming that holding the same number of entries is enough — 3 × 4 and 4 × 3 both hold twelve, yet cannot be combined.

Checking only the rows and forgetting the columns.

Pairing entries that are not in the same position, such as working along a diagonal.

Mishandling a double negative:

3 - (-2) = 5

, not 1.

Writing

B - A

when the question asked for

A - B

Summary

Two matrices can be added or subtracted only when they have the same order — the same number of rows and the same number of columns.
Add or subtract corresponding entries directly: the ones in the same row and the same column.
The resulting matrix keeps the same number of rows and columns as the two you started with.
In symbols, (A ± B) at row i, column j equals the entry of A there plus or minus the entry of B there.
Addition can be written in either order, but subtraction cannot: B − A is the negative of A − B.