Matrices
Basic matrix operations worksheets
Practice six matrix addition, subtraction and scalar multiplication questions with numeric, algebraic or mixed entries. Undefined cases make dimension checks part of the work.
Preview and downloadMatrices practice
Matrix multiplication worksheets
Complete six matrix products with exact numeric, algebraic or mixed entries, and identify products that are undefined because the inner dimensions differ.
Ready-made English PDF · fixed settings
Customization below makes a separate PDF.
Fixed English ready-made files. Custom settings below apply only to the configured PDF.
Ready to print
Start with a ready-made worksheet
Compare the inner dimensions. When they match, multiply each row of the first matrix by each column of the second and add the products.
Download the practice pages alone, or include a separate answer key or teacher model. Longer activities may use more than one page.
Customize this worksheetThe inner dimensions decide
An m × n matrix can multiply an n × p matrix. The shared inner dimension gives the number of products in each row-by-column calculation, while the outer dimensions m × p give the shape of the answer.
Products preserve exact algebraic entries. Undefined cases ask learners to recognize incompatible dimensions before doing unnecessary arithmetic.
Learning objective
What this activity builds
Use row-by-column products to calculate each entry of a matrix product.
Worked example
For A = [[1, 2], [0, 1]] and B = [[1, 0], [3, 1]], calculate AB and BA.
AB = [[7, 2], [3, 1]], while BA = [[1, 2], [3, 7]]. Both are 2 × 2, but they are different.
For example, (AB)₁₁ = 1·1 + 2·3 = 7, while (BA)₁₁ = 1·1 + 0·0 = 1. Matching inner dimensions makes a product defined; it does not make factor order interchangeable.
Teaching notes
Make the practice count
- Write the outer dimensions before calculating.
- Trace one row and one column for each output entry.
- Keep the order of the factors visible because matrix multiplication is order-sensitive.
- Use an exact expression until all products in an entry have been combined.
Matrix multiplication worksheets: dimensions, entries and order
If A is m × n and B is n × p, AB is defined and has dimensions m × p. The entry in row i, column j is the dot product of row i of A and column j of B: multiply corresponding entries, then add. A 2 × 3 matrix times a 3 × 1 column therefore gives a 2 × 1 column; equal full dimensions are not required.
For A = [[1, 2, 0], [−1, 3, 1]] and b = [2, −1, 4]ᵀ, Ab = [1·2 + 2·(−1) + 0·4, (−1)·2 + 3·(−1) + 1·4]ᵀ = [0, −1]ᵀ. The reverse product bA is undefined: its inner dimensions would be 1 and 2. Even when AB and BA both exist, the worked 2 × 2 example shows that they need not be equal.
Each configurable worksheet has six products with numeric, algebraic or mixed entries, including undefined cases. Begin with numeric entries and dimensions checks, then introduce algebraic simplification. Teachers can print Letter or A4 with separate worked answers and ask learners to annotate one row-column calculation per product. Continue to inverses and equations only after multiplication order is secure; this page practices products rather than a complete linear algebra course.
Is multiplying corresponding positions a matrix product?
No. Ordinary matrix multiplication pairs a whole row with a whole column and sums their pairwise products. Entrywise multiplication is a different operation.
What should I write when the inner dimensions do not match?
State that the product is undefined and show the mismatched dimensions. Do not insert zeros or reverse factors to force a result.
Make it yours
Customize the practice
Choose your options, update the preview, then download the exact same pages.
Choose the entry and dimension settings, update the preview and print the same questions with a separate answer key.
Enlarged pages scroll. PDF size stays the same.
A useful next step
Related activities
Matrices
Mixed matrix operations worksheets
Work through six questions that mix addition, subtraction or scalar multiplication with separate matrix products. Exact entries and dimension-aware answers distinguish valid expressions from undefined ones.
Preview and download
Polynomials
Multiplying monomials and polynomials worksheets
Distribute a monomial across a polynomial in ten problems. Select from twelve expression structures, keep powers and signs exact, and check a separate worked answer key.
Preview and download