Practice Solving Systems of Equations with Matrices in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Systems of linear equations can be represented as the matrix equation $Ax = b$ and solved using augmented matrices with row reduction (Gaussian elimination), matrix inverses ( $x = A^{-1}b$ ), or Cramer's rule (using determinants).

Instead of juggling multiple equations with substitution or elimination, pack everything into a matrix and use systematic row operations. It is like organizing a messy desk—once the equations are neatly arranged in a matrix, a mechanical process (row reduction) reveals the answer. Each row operation is an allowed algebraic move (swap equations, scale an equation, add equations) performed on the matrix.

Showing a random 20 of 50 problems.

Example 1

easy

Which row operation swaps two rows?

Example 2

easy

Write

\begin{cases} x = 3 \\ y = 5 \end{cases}

as a solution vector.

Example 3

challenge

Solve the system

\begin{cases} 2x + y = 4 \\ x - y = -1 \end{cases}

three ways agree: give the solution.

Example 4

medium

Use

x = A^{-1}b

to solve

A x = b

where

A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}

and

b = \begin{pmatrix} 4 \\ 11 \end{pmatrix}

Example 5

easy

Solve by inspection:

\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \\ 7 \end{pmatrix}

Example 6

easy

If the coefficient matrix has determinant 0, what does it tell you about the solution?

Example 7

easy

In Cramer's rule for

Ax = b

, what is each variable a ratio of?

Example 8

challenge

Use Cramer's rule to find only

y

\begin{cases} 3x + 2y = 7 \\ x + 4y = 9 \end{cases}

Example 9

hard

Find all

k

for which

\begin{cases} x + 2y = k \\ 2x + 4y = 6 \end{cases}

has infinitely many solutions.

Example 10

easy

What is the augmented matrix of

\begin{cases} 2x - y = 3 \\ x + 4y = 7 \end{cases}

Example 11

medium

Use Cramer's rule to solve

\begin{cases} 2x + y = 5 \\ x + 3y = 10 \end{cases}

Example 12

medium

Find

\det(A)

for

A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}

to confirm a unique solution exists.

Example 13

easy

Write the augmented matrix for

\begin{cases} x + 2y = 4 \\ 3x + y = 5 \end{cases}

Example 14

medium

Solve

\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}

by elimination.

Example 15

easy

Write the coefficient matrix for

\begin{cases} 2x + 3y = 5 \\ x - y = 1 \end{cases}

Example 16

medium

Solve

\begin{cases} 3x - y = 5 \\ 2x + y = 5 \end{cases}

by elimination.

Example 17

medium

What does an augmented matrix row

\left(\begin{array}{cc|c} 0 & 0 & 4 \end{array}\right)

indicate?

Example 18

easy

Find

A^{-1}

for

A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}

Example 19

challenge

Find

\det\!\begin{pmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{pmatrix}

in factored form.

Example 20

easy

Identify the coefficient matrix of

\begin{cases} x + y + z = 1 \\ 2y - z = 0 \\ x + z = 5 \end{cases}

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Related Concepts

Systems of Equations Inverse Matrix Determinant