Matrix Calculator

How This Calculator Works

A matrix is a rectangular grid of numbers that can be added, subtracted, multiplied, or transformed as a single unit, following its own set of algebra rules distinct from ordinary arithmetic. Pick an operation, build Matrix A (and Matrix B, if the operation needs it) using the +/- row and column buttons to resize each grid to whatever dimensions you need, and this calculator applies the matching rule.

Addition and subtraction require both matrices to be exactly the same size. Multiplication requires Matrix A’s column count to match Matrix B’s row count (the result takes Matrix A’s row count and Matrix B’s column count). The determinant only exists for a square matrix (same number of rows and columns). Transpose works on a matrix of any shape, flipping its rows and columns.

This calculator doesn’t attempt row reduction, matrix inverses, or eigenvalues — a genuinely different, much larger undertaking than the operations here.

The Formula

The examples below use 2x2 matrices to illustrate each rule clearly — the same rules extend directly to any size matrix the grid above supports. Let Matrix A have entries $\vA{a_{11}}, \vA{a_{12}}, \vA{a_{21}}, \vA{a_{22}}$ and Matrix B have entries $\vB{b_{11}}, \vB{b_{12}}, \vB{b_{21}}, \vB{b_{22}}$, each written as

$$A = \begin{bmatrix} \vA{a_{11}} & \vA{a_{12}} \\ \vA{a_{21}} & \vA{a_{22}} \end{bmatrix}, \quad B = \begin{bmatrix} \vB{b_{11}} & \vB{b_{12}} \\ \vB{b_{21}} & \vB{b_{22}} \end{bmatrix}$$

• Addition/Subtraction combine matching positions one at a time: $A \pm B = \begin{bmatrix} \vA{a_{11}} \pm \vB{b_{11}} & \vA{a_{12}} \pm \vB{b_{12}} \\ \vA{a_{21}} \pm \vB{b_{21}} & \vA{a_{22}} \pm \vB{b_{22}} \end{bmatrix}$
• Multiplication is NOT element-wise — each entry of the result is a “row times column” dot product: $AB = \begin{bmatrix} \vA{a_{11}}\vB{b_{11}} + \vA{a_{12}}\vB{b_{21}} & \vA{a_{11}}\vB{b_{12}} + \vA{a_{12}}\vB{b_{22}} \\ \vA{a_{21}}\vB{b_{11}} + \vA{a_{22}}\vB{b_{21}} & \vA{a_{21}}\vB{b_{12}} + \vA{a_{22}}\vB{b_{22}} \end{bmatrix}$
• Determinant is a single number describing how much the matrix scales area: $\det(A) = \vA{a_{11}}\vA{a_{22}} - \vA{a_{12}}\vA{a_{21}}$
• Transpose flips the matrix across its main diagonal, swapping the two off-diagonal entries: $A^T = \begin{bmatrix} \vA{a_{11}} & \vA{a_{21}} \\ \vA{a_{12}} & \vA{a_{22}} \end{bmatrix}$

Worked Example

Using $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$:

1. Addition: $A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$.
2. Multiplication: the top-left entry is $1 \times 5 + 2 \times 7 = 19$, the top-right is $1 \times 6 + 2 \times 8 = 22$, the bottom-left is $3 \times 5 + 4 \times 7 = 43$, and the bottom-right is $3 \times 6 + 4 \times 8 = 50$, giving $AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$.
3. Determinant of A: $1 \times 4 - 2 \times 3 = -2$.