MatrixLab

Matrix Calculator (2×2)

The core 2×2 matrix operations together.

Matrix A (2×2)

Matrix B (2×2)

A + B: [[6, 8], [10, 12]]
A × B: [[19, 22], [43, 50]]
det(A): -2
det(B): -2

What you entered

Matrix A: [[1, 2], [3, 4]]Matrix B: [[5, 6], [7, 8]]

1
Add matrices (element-wise)
A + B= [[6, 8], [10, 12]]
2
Multiply matrices (row × column)
A × B= [[19, 22], [43, 50]]
3
Determinant of A
1×4 − 2×3= -2
4
Determinant of B
5×8 − 6×7= -2

Result

A × B: [[19, 22], [43, 50]]

Matrix multiplication combines A and B by pairing each row of A with each column of B — the result represents applying transformation B, then transformation A.

2x2 matrix sum, product and determinants

Addition is entry by entry. Multiplication takes rows of the first against columns of the second, so order matters.

Matrix A	Matrix B	A + B	A x B	det A	det B
[[1,2],[3,4]]	[[5,6],[7,8]]	[[6,8],[10,12]]	[[19,22],[43,50]]	-2	-2
[[2,0],[0,2]]	[[1,1],[1,1]]	[[3,1],[1,3]]	[[2,2],[2,2]]	4	0
[[1,0],[0,1]]	[[3,4],[5,6]]	[[4,4],[5,7]]	[[3,4],[5,6]]	1	-2

The third row multiplies by the identity matrix and the product is unchanged, which is exactly what the identity does - it is the matrix equivalent of multiplying by 1. Note that matrix multiplication is not commutative: A times B generally differs from B times A, which is one of the first places matrix algebra departs from ordinary arithmetic. The determinant of a product equals the product of the determinants, so the first row gives -2 times -2 equals 4 for the product's determinant.

Grids of numbers that transform space

A matrix can represent a system of equations, a rotation, a scaling, or any linear transformation — matrix multiplication combines transformations, applying one after another.

Where matrices are used

Computer graphics (every rotation and zoom on your screen is a matrix multiplication), economics (input-output models), and solving multi-variable equation systems all rely on matrix operations.

Frequently asked questions

How do I add [[1, 2], [3, 4]] and [[5, 6], [7, 8]]?

Add corresponding elements: [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]. Matrix addition is straightforward — just pair up matching positions. The matrix addition calculator breaks this into individual element steps.

Why isn't matrix multiplication just multiplying matching elements?

Element-wise multiplication (Hadamard product) exists but isn't the standard operation because it doesn't compose linear transformations. Standard matrix multiplication uses dot products of rows and columns, which correctly chains transformations — rotate then scale gives the combined transformation matrix.

How is this calculator different from the matrix addition and matrix multiplication ones?

This calculator does both operations together on two 2×2 matrices and also computes both determinants. The dedicated addition and multiplication calculators focus on one operation each with more detailed step-by-step breakdowns.