Matrix Multiplication Calculator (2×2)

About Matrix Multiplication Calculator (2×2)

Matrix multiplication: element (i,j) of the product is the dot product of row i of A and column j of B.

How to use this calculator

1. Enter the four numbers for the first \(2\\times 2\) matrix and the four numbers for the second \(2\\times 2\) matrix.
2. Check that the matrices are both \(2\\times 2\), because this tool is made for that size.
3. Click calculate to multiply the matrices and get the product matrix.
4. Review the displayed result and compare it with the formula if you want to understand each entry.

The formula explained

If the first matrix is \(A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\) and the second is \(B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}\), then the calculator computes \(AB\) by taking row by column products. Each entry of the product matrix is a sum of two products.

• \(a_{11}\) = top left entry of the first matrix
• \(a_{12}\) = top right entry of the first matrix
• \(a_{21}\) = bottom left entry of the first matrix
• \(a_{22}\) = bottom right entry of the first matrix
• \(b_{11}\) = top left entry of the second matrix
• \(b_{12}\) = top right entry of the second matrix
• \(b_{21}\) = bottom left entry of the second matrix
• \(b_{22}\) = bottom right entry of the second matrix

Step by step method

1. Write the first row of \(A\) and the first column of \(B\), then multiply matching entries and add: \(a_{11}b_{11} + a_{12}b_{21}\).
2. Use the first row of \(A\) and the second column of \(B\) for the top right entry: \(a_{11}b_{12} + a_{12}b_{22}\).
3. Repeat with the second row of \(A\) to get the bottom row: \(a_{21}b_{11} + a_{22}b_{21}\) and \(a_{21}b_{12} + a_{22}b_{22}\).

Worked example

Problem. Multiply \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}\).

1. Top left entry: \(1 \cdot 5 + 2 \cdot 7 = 5 + 14 = 19\).
2. Top right entry: \(1 \cdot 6 + 2 \cdot 8 = 6 + 16 = 22\). Bottom left entry: \(3 \cdot 5 + 4 \cdot 7 = 15 + 28 = 43\). Bottom right entry: \(3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50\).
3. So the product is \(AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}\).

Tips and common mistakes

• Matrix multiplication is not done entry by entry, so do not multiply \(a_{11}\) with \(b_{11}\) alone and stop there, you must use a row and a column.
• The order matters, so usually \(AB \\neq BA\). Always keep the first matrix on the left and the second on the right.

Frequently asked questions