Matrix Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

To multiply matrices A (m \times n) and B (n \times p), each entry of the result is the dot product of a row from A with a column from B. The number of columns in A must equal the number of rows in B, and the result is an m \times p matrix.

Imagine each row of A as a question and each column of B as an answer key. You 'grade' each row against each column by multiplying corresponding entries and summing. This is why column count of A must match row count of B—the question and answer key must have the same length.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Matrix multiplication combines rows and columns through dot products. It is NOT commutative: AB \neq BA in general.

Common stuck point: Matrix multiplication is NOT commutative. AB and BA can give completely different results, or one might not even be defined when the other is.

Sense of Study hint: For each entry, run your finger along the row of the first matrix and down the column of the second, multiplying and summing.

Worked Examples

Example 1

medium
Compute \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}.

Solution

  1. 1
    Step 1: Entry (1,1): 1 \cdot 5 + 2 \cdot 7 = 5 + 14 = 19.
  2. 2
    Step 2: Entry (1,2): 1 \cdot 6 + 2 \cdot 8 = 6 + 16 = 22.
  3. 3
    Step 3: Entry (2,1): 3 \cdot 5 + 4 \cdot 7 = 15 + 28 = 43.
  4. 4
    Step 4: Entry (2,2): 3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50.
  5. 5
    Result: \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Answer

\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}
Each entry (i,j) of the product is the dot product of row i of the first matrix with column j of the second matrix. Matrix multiplication is not commutative: AB \neq BA in general.

Example 2

hard
Compute \begin{bmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 4 \\ -1 \end{bmatrix}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Can you multiply a 2 \times 3 matrix by a 2 \times 3 matrix? Why or why not?

Example 2

medium
Compute \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 5 & 7 \end{bmatrix}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

matrix operationsmatrix definition