2x2 Matrix Multiplication Calculator

Reviewed by the CalculatorHub team, edited by James Graham. Built from the row-by-column product rule. Last verified 19 June 2026.

Matrix multiplication combines two matrices by taking the dot product of each row of the first with each column of the second. For two 2x2 matrices the result is another 2x2 matrix with four entries. Order matters: A times B is generally not the same as B times A. Enter the four entries of matrix A and the four entries of matrix B below to see the product A times B with each entry computed.

Matrix A

A row1 col1

A row1 col2

A row2 col1

A row2 col2

Matrix B

B row1 col1

B row1 col2

B row2 col1

B row2 col2

Product row1 col1

0.00

Product row1 col2

0.00

Product row2 col1

0.00

Product row2 col2

0.00

2x2 matrix multiplication formula

A = [[a, b], [c, d]], B = [[e, f], [g, h]]
A * B = [[a*e + b*g, a*f + b*h],
[c*e + d*g, c*f + d*h]]

Each product entry is the dot product of a row of A and a column of B. The row index comes from A and the column index comes from B.

Worked example

With A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]: top-left = 1*5 + 2*7 = 19; top-right = 1*6 + 2*8 = 22; bottom-left = 3*5 + 4*7 = 43; bottom-right = 3*6 + 4*8 = 50. So A times B = [[19, 22], [43, 50]].

2x2 matrix multiplication: frequently asked questions

How do you multiply two 2x2 matrices?

Each entry of the product is the dot product of a row of the first matrix with a column of the second. For the top-left entry, multiply the first row of A by the first column of B and add the two products. Repeat for all four positions: row index from A, column index from B.

Is matrix multiplication commutative?

No. In general A times B does not equal B times A for matrices. The order matters. This calculator computes A times B with A on the left. Swap the two matrices in the inputs to compute B times A.

What is the formula for the product entries?

For A = [[a, b], [c, d]] and B = [[e, f], [g, h]], the product is [[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]]. Each output entry is a sum of two products.

What is the identity matrix for 2x2 multiplication?

The 2x2 identity matrix is [[1, 0], [0, 1]]. Multiplying any 2x2 matrix by the identity, on either side, returns the original matrix unchanged. You can verify this by entering the identity as matrix B.

Sources

NIST Digital Library of Mathematical Functions: dlmf.nist.gov (linear algebra reference).
Standard definition of matrix multiplication (row-by-column dot product).

Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.