Matrix Addition Calculator

This Matrix Addition Calculator helps you add two matrices and shows each step clearly. It works by adding the corresponding entries from matrices of the same dimensions to form a new matrix. This makes it useful for checking answers, understanding how matrix addition works, and practising linear algebra step by step.

Step-by-step method

1. Check that both matrices have the same number of rows and columns.
2. Add entries in the same position, such as row 1 column 1 with row 1 column 1.
3. Write each sum in the matching position of the result matrix.

Formula:

Example 1: Take the matrices below.

Step 1 - Check that both matrices have the same number of rows and columns.

In this problem: Matrix A and Matrix B are both \(2 \times 2\), so they can be added.

Step 2 - Add entries in the same position, such as row 1 column 1 with row 1 column 1.

In this problem: Each entry is paired with the entry in the same row and column.

Step 3 - Write each sum in the matching position of the result matrix.

In this problem: Each sum is placed back into the same position in the answer matrix.

Final answer:

Example 2: Take the matrices below.

Step 1 - Check that both matrices have the same number of rows and columns.

In this problem: Matrix A and Matrix B are both \(3 \times 3\), so they can be added.

Step 2 - Add entries in the same position, such as row 1 column 1 with row 1 column 1.

In this problem: Each entry is paired with the entry in the same row and column.

Step 3 - Write each sum in the matching position of the result matrix.

In this problem: Each sum is placed back into the same position in the answer matrix.

Final answer:

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

1. Check that both matrices have the same number of rows and columns.
2. Add entries in the same position, such as row 1 column 1 with row 1 column 1.
3. Write each sum in the matching position of the result matrix.

Step 1

Step 1 - Check that both matrices have the same number of rows and columns.

In this problem: Matrix A and Matrix B are both \(2 \times 2\), so they can be added.

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

Step 2

Step 2 - Add entries in the same position, such as row 1 column 1 with row 1 column 1.

In this problem: Each entry is paired with the entry in the same row and column.

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

Step 3

Step 3 - Write each sum in the matching position of the result matrix.

In this problem: Each sum is placed back into the same position in the answer matrix.

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

Final Answer

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

See Example 2 Hide Example 2

Problem

\(\begin{pmatrix} 1000 & -2 & 33 \\ 4 & 15 & 6 \\ 7 & 8 & 9 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 30 \\ 4 & 5 & 6 \\ -7 & 0 & 11 \end{pmatrix}\)

Approach

Step-by-step method

1. Check that both matrices have the same number of rows and columns.
2. Add entries in the same position, such as row 1 column 1 with row 1 column 1.
3. Write each sum in the matching position of the result matrix.

Step 1

Step 1 - Check that both matrices have the same number of rows and columns.

In this problem: Matrix A and Matrix B are both \(3 \times 3\), so they can be added.

\(\begin{pmatrix} 1000 & -2 & 33 \\ 4 & 15 & 6 \\ 7 & 8 & 9 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 30 \\ 4 & 5 & 6 \\ -7 & 0 & 11 \end{pmatrix}\)

Step 2

Step 2 - Add entries in the same position, such as row 1 column 1 with row 1 column 1.

In this problem: Each entry is paired with the entry in the same row and column.

\(\begin{pmatrix} 1000 + 1 & -2 + 2 & 33 + 30 \\ 4 + 4 & 15 + 5 & 6 + 6 \\ 7 + (-7) & 8 + 0 & 9 + 11 \end{pmatrix}\)

Step 3

Step 3 - Write each sum in the matching position of the result matrix.

In this problem: Each sum is placed back into the same position in the answer matrix.

\(\begin{pmatrix} 1000 + 1 = 1001 & -2 + 2 = 0 & 33 + 30 = 63 \\ 4 + 4 = 8 & 15 + 5 = 20 & 6 + 6 = 12 \\ 7 + (-7) = 0 & 8 + 0 = 8 & 9 + 11 = 20 \end{pmatrix} = \begin{pmatrix} 1001 & 0 & 63 \\ 8 & 20 & 12 \\ 0 & 8 & 20 \end{pmatrix}\)

Final Answer

\(A + B = \begin{pmatrix} 1001 & 0 & 63 \\ 8 & 20 & 12 \\ 0 & 8 & 20 \end{pmatrix}\)