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 downloadDimensions. Systems. Eigenpairs.
Linear algebra worksheets
Build a computational foundation with exact matrices and linear systems. Follow the sequence below, or choose the step that matches your course.
Before you start
Be comfortable with signed fractions, algebraic expressions, simultaneous linear equations and quadratic factoring. Calculus is not required for this path.
This is a bounded computational introduction: matrix arithmetic, 2 × 2 inverse calculations, three-variable row reduction and designed 2 × 2 eigenproblems. It is not a complete university linear algebra course. Abstract vector spaces, proof-based span and independence, general linear transformations, orthogonality and spectral theory need additional teaching and resources.
1. Read dimensions and operate entry by entry
State rows × columns. Add or subtract only equal-sized matrices and distribute a scalar to every entry.
2. Multiply rows by columns
Match inner dimensions and predict the outer dimensions first. Keep the factor order: AB and BA may differ, or one may be undefined.
Matrices
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.
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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 download3. Use inverses to solve matrix equations
For a 2 × 2 matrix, test ad − bc before using the inverse formula. To solve AX = B, multiply on the left by A⁻¹ and check AX in the original equation.
Matrices
Matrix inverse worksheets
Decide whether six numeric 2 × 2 matrices have inverses, then find six exact inverses or state that no inverse exists.
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Matrices
Matrix equations worksheets
Solve six exact matrix equations by isolating a scalar multiple of X or multiplying by a 2 × 2 inverse on the left.
Preview and download4. Interpret Gaussian elimination and RREF
Track elementary row operations in exact three-variable systems. Identify pivot columns, free variables and contradiction rows; verify unique or parametric answers by substitution.
Advanced math
Gaussian Elimination and RREF worksheets
Row-reduce exact 3-variable systems with unique, infinite, and inconsistent outcomes and give parametric answers.
Preview and download5. Find and check 2 × 2 eigenpairs
Use det(A − λI) = 0 and solve (A − λI)v = 0. An eigenvector must be nonzero; a repeated eigenvalue need not provide two independent eigenvectors.
Advanced math
2 by 2 Eigenvalues and Eigenvectors worksheets
Solve exact designed 2 by 2 cases with distinct or repeated eigenvalues and valid eigenspace bases.
Preview and downloadWorked system: calculate and check
Solve Ax = b for A = [[1, 2], [0, 1]] and b = [5, 2]ᵀ.
The second row gives x₂ = 2. The first gives x₁ + 2(2) = 5, so x₁ = 1. Thus x = [1, 2]ᵀ.
A⁻¹ = [[1, −2], [0, 1]], giving A⁻¹b = [1, 2]ᵀ. Multiplying back gives Ax = [1 + 4, 2]ᵀ = b. These are two consistent methods, not permission to reverse matrix order.
Extend the check to an eigenpair
For A = [[2, 1], [0, 3]], det(A − λI) = (2 − λ)(3 − λ). For λ = 2, v = (1, 0) works; for λ = 3, v = (1, 1) works. Check A(1, 1)ᵀ = (3, 3)ᵀ = 3(1, 1)ᵀ. Scalar multiples give the same eigendirection, and zero is excluded.
Prepare an editable review packet
Introductory linear algebra review packet
Progress from matrix arithmetic and multiplication through inverses and equations to RREF and designed 2 × 2 eigenproblems.
Begin with 6 complete worksheets. Remove or reorder them, then choose Letter or A4 and student or teacher editions.
Edit this packet →For teachers and tutors
Start with a short dimensions check before assigning products. Use the ordered packet for review, or remove steps not yet taught. Ask learners to show row operations, label any free parameters and verify a proposed solution or eigenvector in the original equation; the separate worked key supports marking and discussion.
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