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Matrix products

Multiply matrices row by column

Set the dimensions of A and B, enter each value, and calculate AB. The calculator checks compatibility and shows how every result entry is formed.

Enter matrix A and matrix B

Matrix A

2 × 3

Matrix B

3 × 2

Compatible: columns(A) = 3 and rows(B) = 3. The product will be 2 × 2.

Try an example

Result

The product AB is a 2 by 2 matrix

[5864139154]\begin{bmatrix}58 & 64 \\ 139 & 154\end{bmatrix}
Conditions
  • A has dimensions 2 by 3 and B has dimensions 3 by 2.
  • The inner dimensions match: columns(A) = 3 = rows(B).

Steps

  1. Check the dimensions A is 2 by 3 and B is 3 by 2, so AB exists and will be 2 by 2.(2×3)(3×2)⟶2×2(2\times3)(3\times2)\longrightarrow2\times2
  2. Calculate the first result entry Multiply row 1 of A by column 1 of B and add the products.c11=1(7)+2(9)+3(11)=58c_{11}=1(7)+2(9)+3(11)=58
  3. Repeat for every row and column pair Each output position uses one row of A and one column of B.AB=[5864139154]AB=\begin{bmatrix}58&64\\139&154\end{bmatrix}
Independent check

A separate finite-number calculation reproduced all four entries and confirmed the 2 by 2 output shape.

Scope

What this matrix multiplication covers

The product AB exists when the number of columns in A equals the number of rows in B. If A is m by n and B is n by p, then AB is m by p.

Check whether AB is defined

Compare columns of A with rows of B before multiplying, and preserve the requested dimensions when they do not match.

Examples: 2 by 3 times 3 by 4 is defined, 2 by 3 times 2 by 2 is undefined

Multiply rectangular matrices

Use independent dimensions for A and B, then form an output with rows of A and columns of B.

Examples: 2 by 3 times 3 by 2, 1 by 4 times 4 by 3

Inspect every dot product

See the row entries, column entries, products, and sum that produce each output cell.

Examples: c11, c12, c21, c22

Keep fractions exact

Multiply and add rational entries without rounding away the structure of the answer.

Examples: 1/2, -3/4, 1.25

How to use it

Enter enough information for one clear task

  1. 1
    Choose the size of A

    Set rows and columns from one to four, then enter values in row order. Existing values stay in place when a dimension is changed.

  2. 2
    Choose the size of B

    For AB to exist, the row count of B must equal the column count of A. The compatibility message updates before calculation.

  3. 3
    Multiply matrices

    The result has one row for each row of A and one column for each column of B.

  4. 4
    Open the cell-level steps

    For each cij, follow row i of A across column j of B. The displayed sum explains the exact result entry.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Multiply two 2 by 2 matrices

Use each row of A against each column of B.

[2−103][452−2]\begin{bmatrix}2&-1\\0&3\end{bmatrix}\begin{bmatrix}4&5\\2&-2\end{bmatrix}

Expected result

[6126−6]\begin{bmatrix}6&12\\6&-6\end{bmatrix}

Rectangular product

The inner dimensions are both 3, so the result is 2 by 2.

[123456][789101112]\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}\begin{bmatrix}7&8\\9&10\\11&12\end{bmatrix}

Expected result

[5864139154]\begin{bmatrix}58&64\\139&154\end{bmatrix}

Row times column

One dot product produces a 1 by 1 result.

[2−14][35−2]\begin{bmatrix}2&-1&4\end{bmatrix}\begin{bmatrix}3\\5\\-2\end{bmatrix}

Expected result

[−7]\begin{bmatrix}-7\end{bmatrix}

Multiply by the identity matrix

The identity columns select the original columns of A.

[3−251][1001]\begin{bmatrix}3&-2\\5&1\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}

Expected result

[3−251]\begin{bmatrix}3&-2\\5&1\end{bmatrix}

Multiply by a zero matrix

Every dot product contains only zero contributions.

[1−423][0000]\begin{bmatrix}1&-4\\2&3\end{bmatrix}\begin{bmatrix}0&0\\0&0\end{bmatrix}

Expected result

[0000]\begin{bmatrix}0&0\\0&0\end{bmatrix}

Multiply exact fractions

Keep each rational product exact before adding.

[12102][20112]\begin{bmatrix}\frac12&1\\0&2\end{bmatrix}\begin{bmatrix}2&0\\1&\frac12\end{bmatrix}

Expected result

[21221]\begin{bmatrix}2&\frac12\\2&1\end{bmatrix}

See why order matters

AB and BA use different row-column pairings.

A=[1101],B=[1011]A=\begin{bmatrix}1&1\\0&1\end{bmatrix},\quad B=\begin{bmatrix}1&0\\1&1\end{bmatrix}

Expected result

AB=[2111]≠[1112]=BAAB=\begin{bmatrix}2&1\\1&1\end{bmatrix}\ne\begin{bmatrix}1&1\\1&2\end{bmatrix}=BA

Complete example

Multiply a 2 by 3 matrix by a 3 by 2 matrix

The three columns of A match the three rows of B. The product will use the two outer dimensions, so AB is 2 by 2.

A=[123456],B=[789101112]A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix},\qquad B=\begin{bmatrix}7&8\\9&10\\11&12\end{bmatrix}
  1. 1
    Calculate the first row of the product

    Use row 1 of A with columns 1 and 2 of B.

    c11=1(7)+2(9)+3(11)=58,c12=1(8)+2(10)+3(12)=64c_{11}=1(7)+2(9)+3(11)=58,\qquad c_{12}=1(8)+2(10)+3(12)=64
  2. 2
    Calculate the second row of the product

    Use row 2 of A with the same two columns of B.

    c21=4(7)+5(9)+6(11)=139,c22=4(8)+5(10)+6(12)=154c_{21}=4(7)+5(9)+6(11)=139,\qquad c_{22}=4(8)+5(10)+6(12)=154
  3. 3
    Place each value in its row and column

    The four dot products fill a 2 by 2 result matrix.

    AB=[5864139154]AB=\begin{bmatrix}58&64\\139&154\end{bmatrix}
AB=[5864139154]AB=\begin{bmatrix}58&64\\139&154\end{bmatrix}

Verification: Independent numerical loops reproduce 58, 64, 139, and 154, and the result has rows(A)=2 and columns(B)=2.

Avoidable errors

Common mistakes and how to fix them

Comparing the outer dimensions

Problem: Require rows of A to equal columns of B.

Why it matters: The dot products pair a row of A with a column of B, so their lengths come from columns of A and rows of B.

Better approach: Check columns(A)=rows(B). Use rows(A) by columns(B) for the output shape.

Multiplying corresponding entries

Problem: Multiply a11 by b11, a12 by b12, and stop.

Why it matters: That is an entrywise operation, not the row-by-column definition of matrix multiplication.

Better approach: For cij, multiply every entry in row i of A by the matching entry in column j of B, then add.

Assuming AB equals BA

Problem: Reverse the matrices without recalculating.

Why it matters: The dimensions may change or BA may be undefined, and even two defined products usually have different entries.

Better approach: Treat order as part of the problem and calculate the requested product only.

Predicting the wrong result size

Problem: Copy the dimensions of A or B automatically.

Why it matters: The shared inner dimension disappears into each dot product.

Better approach: Keep the two outer dimensions: rows of A by columns of B.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Every entry is parsed by the site's Nerdamer mathematics library before multiplication.
  • Each result cell is built from an explicit row-by-column exact sum.
  • A separate finite-number implementation recomputes every product cell.
  • The result shape is checked against rows(A) by columns(B).

When to stop and revise the input

  • This release accepts finite real integers, decimals, and simple fractions, not variables or complex entries.
  • Each input matrix is limited to 4 by 4 so cell-level explanations remain usable.
  • The calculator evaluates AB only; chained products and powers require separate steps.
  • An incompatible product remains undefined and is never changed into entrywise multiplication.

Common questions

Multiply matrices row by column FAQ

When can matrix A be multiplied by matrix B?

AB is defined when the number of columns in A equals the number of rows in B. Those are the inner dimensions used in every dot product.

What size is the product matrix?

If A is m by n and B is n by p, then AB is m by p. The result keeps the rows of A and the columns of B.

Why does the calculator show one step per result cell?

Each cij comes from a different row-column pairing. Showing the products and their sum makes it possible to locate a sign or arithmetic mistake instead of accepting an opaque result.

Does AB always equal BA?

No. Matrix multiplication is generally not commutative. BA can have different entries, a different shape, or be undefined even when AB exists.

Can matrix entries be fractions?

Yes. Enter simple fractions such as 1/2 or -3/4. The calculator keeps them exact through the row-by-column sums.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.