Solving Systems of Equations with Matrices Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Systems of Equations with Matrices.

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

Concept 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.

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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: Write the system as a matrix equation and solve by row reduction, $x=A^{-1}b$ , or Cramer's rule.

Common stuck point: The procedure for solving systems of equations with matrices is the easy part; the trap is using $x=A^{-1}b$ when $\det A=0$ . Asking "Can I write the system as $Ax=b$ and solve mechanically by row ops, inverse, or determinants?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I write the system as $Ax=b$ and solve mechanically by row ops, inverse, or determinants?

Worked Examples

Example 1

medium

Solve using the inverse matrix:

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

Answer

x = 2, y = 1

First step

Step 1: Write as

Ax = b

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

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Example 2

hard

Use Cramer's rule to solve:

\begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}

Example 3

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 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

hard

Solve using row reduction:

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

Practice Problems

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

Example 1

easy

Write the system

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

as a matrix equation

Ax = b

Example 2

medium

Solve using the inverse:

\begin{cases} x + 2y = 8 \\ 3x + 5y = 19 \end{cases}

Example 3

easy

Write the coefficient matrix for

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

Example 4

easy

Write the augmented matrix for

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

Example 5

easy

Ax = b

, what does the vector

x

represent?

Example 6

easy

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

Example 7

easy

Write

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

as a solution vector.

Example 8

easy

How many equations and variables does a

3 \times 3

coefficient matrix represent?

Example 9

easy

What row operation scales row 1 of

\left(\begin{array}{cc|c} 2 & 4 & 6 \\ 1 & 1 & 2 \end{array}\right)

to make its leading entry 1?

Example 10

easy

In Cramer's rule for

Ax = b

, what is each variable a ratio of?

Example 11

medium

Solve

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

by elimination.

Example 12

medium

Use Cramer's rule to solve

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

Example 13

medium

Row reduce

\left(\begin{array}{cc|c} 1 & 2 & 4 \\ 2 & 5 & 9 \end{array}\right)

and solve.

Example 14

medium

Solve using

x = A^{-1}b

for

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

b = \begin{pmatrix} 3 \\ 5 \end{pmatrix}

Example 15

medium

Does

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

have a solution? Use the determinant.

Example 16

medium

Solve

\begin{cases} x + y + z = 6 \\ y + z = 5 \\ z = 3 \end{cases}

by back-substitution.

Example 17

medium

What does an augmented matrix row

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

indicate?

Example 18

medium

Solve

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

by elimination.

Example 19

medium

Find

\det(A)

for

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

to confirm a unique solution exists.

Example 20

challenge

Solve the system

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

three ways agree: give the solution.

Example 21

challenge

For what

k

does

\begin{cases} x + 2y = 3 \\ 2x + ky = 6 \end{cases}

have infinitely many solutions?

Example 22

challenge

Use Cramer's rule to find only

y

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

Example 23

easy

Compute

\det\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}

Example 24

easy

What is the augmented matrix of

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

Example 25

easy

Identify the coefficient matrix of

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

Example 26

easy

Find

A^{-1}

for

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

Example 27

easy

Compute

\det\begin{pmatrix} 4 & 6 \\ 2 & 3 \end{pmatrix}

Example 28

medium

Solve

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

by elimination.

Example 29

medium

Find

A^{-1}

for

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

Example 30

medium

Use Cramer's rule to solve

\begin{cases} 4x + y = 6 \\ x + 2y = 5 \end{cases}

Example 31

medium

Row-reduce

\left(\begin{array}{cc|c} 1 & -2 & 3 \\ 2 & -3 & 5 \end{array}\right)

to find

(x, y)

Example 32

medium

Compute

\det\begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{pmatrix}

Example 33

medium

For what

k

\begin{pmatrix} 1 & 2 \\ 3 & k \end{pmatrix}

singular?

Example 34

hard

Solve

\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases}

Example 35

hard

Compute

\det\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}

Example 36

hard

Use Cramer's rule to find only

x

\begin{cases} 2x + y - z = 3 \\ x - y + z = 0 \\ x + 2y + z = 6 \end{cases}

Example 37

hard

Find the inverse of

A = \begin{pmatrix} 1 & 2 \\ 3 & 7 \end{pmatrix}

Example 38

hard

Find all

k

for which

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

has infinitely many solutions.

Example 39

hard

Compute

A^{-1}b

for

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

and

b = \begin{pmatrix} 7 \\ 4 \end{pmatrix}

Example 40

hard

Show that

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

has no solution using the augmented matrix.

Example 41

challenge

Find

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

in factored form.

Example 42

challenge

A

is invertible and

\det A = 4

, what is

\det(2A^{-1})

for a

3\times 3

matrix?

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Related Concepts

Systems of Equations Inverse Matrix Determinant

Background Knowledge

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

systems of equationsinverse matrixdeterminant