Matrix Multiplication Calculator

1. Choose the matrix size, 2 by 2 or 3 by 3.
2. Enter Matrix A row by row, separated by commas.
3. Enter Matrix B the same way.
4. Read the product, shown row by row, and its trace.

Here is what each result means:

Matrix multiplication combines two matrices into a new one using the row-by-column rule. Each entry of the product is a row of the first matrix multiplied term by term with a column of the second, and those products added. So the entry in row i, column j comes from row i of A and column j of B.

Matrix multiplication is how linear transformations combine, so it powers computer graphics, rotations, systems of equations and machine learning. Unlike ordinary multiplication, it is not commutative: A times B is usually different from B times A, and the order matters.

It computes each entry of the product as the dot product of a row and a column.

1. Take a row of A and a column of B.
2. Multiply matching entries and add them for one product entry.
3. Repeat for every row and column to fill the product matrix.

The determinant of the product equals the product of the determinants, which you can check with the determinant calculator.

Multiply the 2x2 matrices A with rows 1, 2 and 3, 4, and B with rows 5, 6 and 7, 8.

Calculation: the top-left entry is the first row of A dotted with the first column of B: 1 x 5 + 2 x 7 = 19. Continuing gives the product [19, 22 ; 43, 50]. The trace, 19 plus 50, is 69.

Each entry pairs one row of A with one column of B.

For two matrices to multiply, the columns of the first must match the rows of the second. Here both are square and the same size, so they always fit.

The two operations combine matrices in very different ways.

Addition needs matrices of the same shape and is commutative. Multiplication combines rows and columns and is not, so A times B and B times A can differ.

The Order of the Matrices

Multiplication is not commutative, so swapping A and B usually changes the result.

The Identity Matrix

Multiplying by the identity, ones on the diagonal, leaves a matrix unchanged.

A Zero Row or Column

Zeros in A or B carry through, producing zero entries in the product.

Linear Algebra Homework

Multiply matrices quickly and check each entry.

Transformations

Combine rotations, scalings or reflections by multiplying their matrices.

Systems and Models

Apply a matrix to data, as in graphics and machine learning.

1. Multiplying Entry by Entry

Matrix multiplication is not element-wise. Each entry is a row dotted with a column.

2. Assuming a B Equals B A

Order matters. The two products are usually different matrices.

3. Mismatched Sizes

The columns of the first must equal the rows of the second. Here both are the same square size.

4. Losing Track of Rows and Columns

Enter each matrix row by row, left to right, so the entries land in the right places.

5. Skipping a Term in the Sum

Each product entry adds one term per dimension. Missing one gives a wrong number.

The multiplication is exact; only the displayed decimals are rounded.

Use the row-by-column rule: each entry of the product is a row of the first matrix multiplied term by term with a column of the second, and the products added. This tool does that for every entry.

No. In general A times B is not the same as B times A, and often only one order is even defined. The order of the matrices matters.

The number of columns in the first matrix must equal the number of rows in the second. This tool handles two square matrices of the same size, 2x2 or 3x3, so they always fit.

It is a square matrix with ones on the diagonal and zeros elsewhere. Multiplying any matrix by the identity leaves it unchanged, like multiplying a number by 1.

The trace is the sum of the diagonal entries, from the top left to the bottom right. It is a quick summary value and appears in many linear-algebra results.

It combines linear transformations, so it drives computer graphics, rotations and scaling, solving systems of equations, and the layers of a neural network.

Element-wise multiplication is a different operation. Matrix multiplication mixes rows and columns so that it represents composing two transformations, which element-wise multiplication does not.

A matrix of all zeros. Every product entry sums terms that each include a zero, so the whole product is zero.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Matrix multiplication (Wikipedia).
• How to multiply matrices (Maths Is Fun).
• Trace (linear algebra) (Wikipedia).