Matrix Addition Notes

Published on: September 29, 2026

This Matrix Addition Calculator adds two matrices and shows the working line by line. Two matrices can only be added when they have the same number of rows and columns, because every entry needs a partner sitting in the same position. Each pair is added on its own and the total goes straight back into that position, so the answer is the same size as the two it came from. Following the same pattern every time makes matrix addition easy to check and repeat.

Step-by-step method

Set up the problem.
Write the formula.
Pair the entries in matching positions.
Add each pair and write the total in its place.

Formula:

\(\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}]\text{, then } A + B = [a_{ij} + b_{ij}]\)

See Example 1 Hide Example 1

Problem

\(\begin{pmatrix} 12 & -3 \\ 40 & 7 \end{pmatrix} + \begin{pmatrix} 5 & 18 \\ -9 & 2 \end{pmatrix}\)

Approach

Step-by-step method

Set up the problem.
Write the formula.
Pair the entries in matching positions.
Add each pair and write the total in its place.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A and Matrix B are both \(2 \times 2\), so every entry in one has a partner in the same position in the other.

\(\begin{pmatrix} 12 & -3 \\ 40 & 7 \end{pmatrix} + \begin{pmatrix} 5 & 18 \\ -9 & 2 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Two matrices of the same size line up entry for entry, so they are added one position at a time rather than as whole blocks.

\(\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}]\text{, then } A + B = [a_{ij} + b_{ij}]\)

Step 3

Step 3 - Pair the entries in matching positions.

In this problem: Pairing the entries in matching positions gives \(12 + 5\), \(-3 + 18\), \(40 + (-9)\), \(7 + 2\).

\(\begin{pmatrix} 12 + 5 & -3 + 18 \\ 40 + (-9) & 7 + 2 \end{pmatrix}\)

Step 4

Step 4 - Add each pair and write the total in its place.

In this problem: Working each position out: \(12 + 5 = 17\), \(-3 + 18 = 15\), \(40 + (-9) = 31\), \(7 + 2 = 9\).

\(\begin{pmatrix} 12 + 5 & -3 + 18 \\ 40 + (-9) & 7 + 2 \end{pmatrix} = \begin{pmatrix} 17 & 15 \\ 31 & 9 \end{pmatrix}\)

Final Answer

\(A + B = \begin{pmatrix} 17 & 15 \\ 31 & 9 \end{pmatrix}\)

See Example 2 Hide Example 2

Problem

\(\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} + \begin{pmatrix} 7 & 8 & 9 \\ 1 & 0 & 2 \end{pmatrix}\)

Approach

Step-by-step method

Set up the problem.
Write the formula.
Pair the entries in matching positions.
Add each pair and write the total in its place.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A and Matrix B are both \(2 \times 3\), so every entry in one has a partner in the same position in the other.

\(\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} + \begin{pmatrix} 7 & 8 & 9 \\ 1 & 0 & 2 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Two matrices of the same size line up entry for entry, so they are added one position at a time rather than as whole blocks.

\(\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}]\text{, then } A + B = [a_{ij} + b_{ij}]\)

Step 3

Step 3 - Pair the entries in matching positions.

In this problem: Pairing the entries in matching positions gives \(1 + 7\), \(2 + 8\), \(3 + 9\), \(4 + 1\), \(5 + 0\), \(6 + 2\).

\(\begin{pmatrix} 1 + 7 & 2 + 8 & 3 + 9 \\ 4 + 1 & 5 + 0 & 6 + 2 \end{pmatrix}\)

Step 4

Step 4 - Add each pair and write the total in its place.

In this problem: Working each position out: \(1 + 7 = 8\), \(2 + 8 = 10\), \(3 + 9 = 12\), \(4 + 1 = 5\), \(5 + 0 = 5\), \(6 + 2 = 8\).

\(\begin{pmatrix} 1 + 7 & 2 + 8 & 3 + 9 \\ 4 + 1 & 5 + 0 & 6 + 2 \end{pmatrix} = \begin{pmatrix} 8 & 10 & 12 \\ 5 & 5 & 8 \end{pmatrix}\)

Final Answer

\(A + B = \begin{pmatrix} 8 & 10 & 12 \\ 5 & 5 & 8 \end{pmatrix}\)

See Example 3 Hide Example 3

Problem

\(\begin{pmatrix} 3 & 8 \\ -2 & 5 \end{pmatrix} + \begin{pmatrix} 6 & -4 \\ 7 & 1 \end{pmatrix}\)

Approach

Step-by-step method

Set up the problem.
Write the formula.
Pair the entries in matching positions.
Add each pair and write the total in its place.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A and Matrix B are both \(2 \times 2\), so every entry in one has a partner in the same position in the other.

\(\begin{pmatrix} 3 & 8 \\ -2 & 5 \end{pmatrix} + \begin{pmatrix} 6 & -4 \\ 7 & 1 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Two matrices of the same size line up entry for entry, so they are added one position at a time rather than as whole blocks.

\(\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}]\text{, then } A + B = [a_{ij} + b_{ij}]\)

Step 3

Step 3 - Pair the entries in matching positions.

In this problem: Pairing the entries in matching positions gives \(3 + 6\), \(8 + (-4)\), \(-2 + 7\), \(5 + 1\).

\(\begin{pmatrix} 3 + 6 & 8 + (-4) \\ -2 + 7 & 5 + 1 \end{pmatrix}\)

Step 4

Step 4 - Add each pair and write the total in its place.

In this problem: Working each position out: \(3 + 6 = 9\), \(8 + (-4) = 4\), \(-2 + 7 = 5\), \(5 + 1 = 6\).

\(\begin{pmatrix} 3 + 6 & 8 + (-4) \\ -2 + 7 & 5 + 1 \end{pmatrix} = \begin{pmatrix} 9 & 4 \\ 5 & 6 \end{pmatrix}\)

Final Answer

\(A + B = \begin{pmatrix} 9 & 4 \\ 5 & 6 \end{pmatrix}\)

See Example 4 Hide Example 4

Problem

\(\begin{pmatrix} 2 & 0 & 5 \\ -1 & 3 & 4 \end{pmatrix} + \begin{pmatrix} 4 & 6 & -2 \\ 3 & 1 & 0 \end{pmatrix}\)

Approach

Step-by-step method

Set up the problem.
Write the formula.
Pair the entries in matching positions.
Add each pair and write the total in its place.

Step 1

Step 1 - Set up the problem.

In this problem: Matrix A and Matrix B are both \(2 \times 3\), so every entry in one has a partner in the same position in the other.

\(\begin{pmatrix} 2 & 0 & 5 \\ -1 & 3 & 4 \end{pmatrix} + \begin{pmatrix} 4 & 6 & -2 \\ 3 & 1 & 0 \end{pmatrix}\)

Step 2

Step 2 - Write the formula.

In this problem: Two matrices of the same size line up entry for entry, so they are added one position at a time rather than as whole blocks.

\(\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}]\text{, then } A + B = [a_{ij} + b_{ij}]\)

Step 3

Step 3 - Pair the entries in matching positions.

In this problem: Pairing the entries in matching positions gives \(2 + 4\), \(0 + 6\), \(5 + (-2)\), \(-1 + 3\), \(3 + 1\), \(4 + 0\).

\(\begin{pmatrix} 2 + 4 & 0 + 6 & 5 + (-2) \\ -1 + 3 & 3 + 1 & 4 + 0 \end{pmatrix}\)

Step 4

Step 4 - Add each pair and write the total in its place.

In this problem: Working each position out: \(2 + 4 = 6\), \(0 + 6 = 6\), \(5 + (-2) = 3\), \(-1 + 3 = 2\), \(3 + 1 = 4\), \(4 + 0 = 4\).

\(\begin{pmatrix} 2 + 4 & 0 + 6 & 5 + (-2) \\ -1 + 3 & 3 + 1 & 4 + 0 \end{pmatrix} = \begin{pmatrix} 6 & 6 & 3 \\ 2 & 4 & 4 \end{pmatrix}\)

Final Answer

\(A + B = \begin{pmatrix} 6 & 6 & 3 \\ 2 & 4 & 4 \end{pmatrix}\)