Skill: Determine the dimensions of a matrix product

Explanation and Free Practice Resources

For finite matrices, the product of an $m \times n$ matrix and an $r \times p$ matrix is defined only when the inner dimensions agree, $n=r$ ; the resulting matrix has dimensions $m \times p$ . The shared dimension governs whether multiplication is possible, while the outer dimensions determine the product’s size, independently of the matrix entries; block-matrix and other generalized product settings are not included.

On this page:

Learn “Determine the dimensions of a matrix product” with a detailed explanation of the concept
Master “Determine the dimensions of a matrix product” with unlimited, free practice problems
Download (or modify) a free worksheet on Determine the dimensions of a matrix product you can use at home or in class with practice problems that scaffold and explore “Determine the dimensions of a matrix product” from different perspectives.
Start working on adaptive units like “Matrices - Multiplication - Intro” that introduce, teach, or practice “Determine the dimensions of a matrix product” to maximize your learning.

Detailed Explanation: Determine the dimensions of a matrix product

To determine the dimensions of a matrix product, compare the inner dimensions. The product is defined only if they match. If they do, the outer dimensions give the size of the result.

For example, suppose $A$ is a $3 \times 4$ matrix and $B$ is a $4 \times 2$ matrix. In the product $AB$ :

Write the dimensions in order: $(3 \times 4)(4 \times 2)$ .
Check the inner dimensions: $4$ and $4$ match, so the product is defined.
Read the outer dimensions: $3$ and $2$ .

Therefore, $AB$ has dimensions $3 \times 2$ . The entries of the matrices do not affect this dimension check.

Learn by doing: Determine the dimensions of a matrix product

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Your results

Practice with unlimited practice problems

Matrices - Multiply - Two Matrices to Product Dimensions

Earned ?

Download/Modify Free Grade 12 'Determine the dimensions of a matrix product' Worksheet

Print this free 'Determine the dimensions of a matrix product' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.

Units Teaching This Skill

These adaptive units introduce, teach, or reinforce this skill (Determine the dimensions of a matrix product) through scaffolded problems that grow in difficulty as concepts are understood.