How to Multiply Matrices Step by Step

How does matrix multiplication work, step by step?

Leave the operation on Multiply and fill both boxes. One row goes on each line. Numbers on a row are separated by spaces or commas. The default pair is 1 2 / 3 4 times 5 6 / 7 8. The product is 19 22 on the first row and 43 50 on the second.

The headline is the top-left entry, 19. Step 1 shows where it came from: row 1 of A dotted with column 1 of B, written 1×5 + 2×7 = 19. The other entries are listed as rows under that step. You get one worked step, then the whole matrix, not a locked expansion.

Columns of A must equal rows of B. A 2×3 times a 2×3 fails. The message names both shapes. It does not say only that the product is undefined. The arithmetic stays in the browser. Nothing is uploaded.

How do you enter a matrix?

Each line is a row. 1 2 3 and then 4 5 6 is a 2×3 matrix. Commas work the same way, so 1, 2, 3 is one row of three. A semicolon starts a row too. 1, 2; 3, 4 is the 2×2 default written on one line. Blank lines are dropped.

Every row needs the same count of numbers. If row 2 has two numbers and row 1 has three, the page says so and stops. A value that is not a number is named by its row. The largest grid is 8 by 8. A wider paste is refused before multiply or add runs.

Matrix B is required for add, subtract, and multiply. Transpose, determinant, and inverse read A only. You can leave B as it is. Those three operations ignore it. Clearing B matters only when you switch back to an operation that needs two matrices.

When do the shapes have to match?

Add and subtract need the same shape on both sides. A 2×3 plus a 2×2 is refused, and the message reads A is 2×3 and B is 2×2. Multiply is stricter about a different pair of numbers. The columns of A must equal the rows of B. The result then has the rows of A and the columns of B.

So a 2×3 times a 3×2 is a 2×2. A 2×3 times a 2×3 is not a product. The inner 3 and 2 disagree. People meet this when both matrices were typed in the same orientation. Transpose one of them and try again. The error is the orientation, not a broken formula.

The check runs before any dot product. You do not get a partial matrix with a hole where the sizes failed. Either the shapes are legal and every entry is filled, or you get the sentence that names both shapes. That sentence is the whole diagnosis.

What do the first steps show?

The first step is one entry, written out. For a product it is the first dot product. For a sum it is the first pair added, such as 1 + 5 = 6 if you switch the default matrices to Add. For a difference it is the same pair subtracted. You can check that line by hand, then trust the remaining rows.

A 2 by 2 determinant shows the textbook line. 1×4 - 2×3 = -2 for the default matrix. Larger square matrices report the value from elimination instead of a wall of minors. An inverse reports the top-left entry after Gauss-Jordan on the augmented matrix. The row list is the rest of the inverse.

This is not a full tableau and not reduced row echelon form. There is no RREF operation. If a course asks you to show every row operation, write those lines yourself from the one step and the finished matrix. The page will not pretend a single dot product is the entire proof.

How do determinant and inverse differ?

The determinant is one number. The inverse is another matrix. Both need a square A. A 2×3 has neither. The message names that shape and stops. For a 2×2, determinant is a×d - b×c. The default gives -2, which is not zero, so the inverse exists. To compute a 2 by 2 inverse by hand, swap a and d, negate b and c, and divide every entry by the determinant.

The inverse of 1 2 / 3 4 has top-left entry -2, then 1 on the rest of the first row, and 1.5 and -0.5 on the second. Multiply that inverse by the original and you should get the identity, 1 0 / 0 1. Switch back to Multiply and paste the inverse into B if you want to see that product.

Determinant 0 means no inverse. 1 2 / 2 4 is the example: the second row is twice the first, and the page says the determinant is 0, so this matrix has no inverse. It does not print a matrix of zeros or of infinities. Zero determinant is a stop, not a result grid.

Why does 0.1 plus 0.2 print as 0.3?

Binary floating point cannot store 0.1 exactly. Adding 0.1 and 0.2 produces a value that prints as 0.30000000000000004 if you show every bit. A matrix of tenths would then look unequal to the pencil answer even when the arithmetic is right. This page tidies each entry.

A value that is within a hair of a whole number prints as that whole number. A tenth prints as 0.3, not as a long tail. The assumption under the result says this in one line. The tidy step does not change a real difference, such as 19 versus 19.5. It only removes noise.

Fractions you want to keep exact, such as 1/3, should be entered as a decimal you accept, or reduced on the fraction calculator first. This page does not do fraction arithmetic. It multiplies the decimals you typed, then tidies the printout.

Which operation should you pick?

Multiply when a textbook says product or composition. Add when it says sum. Subtract when it says difference. Transpose when it says flip the rows into columns, because a transpose swaps rows and columns. Determinant when it asks for one number from a square matrix. Inverse when it asks for the matrix that multiplies back to the identity.

Read the shape note on the result. A 2×2 result from a 2×3 times a 3×2 is the legal size. If you expected a 2×3, the operation or the orientation is wrong, even when no error appeared. The rows under the headline are the check. Count them.

Then change one entry and look again. The step and the rows update together. That loop is the reason to use the page instead of a single chat reply: the grid stays put, and the shape sentence appears only when the sizes actually disagree. Runs entirely on your device. Nothing is uploaded.

For example, multiply the defaults. Input: 1 2 / 3 4 times 5 6 / 7 8. Output: 19 22 on the first row and 43 50 on the second. Switch to Inverse and the top-left entry for 1 2 / 3 4 is -2.