Matrix Definition Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Definition.

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

Concept Recap

A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical). An $m \times n$ matrix has $m$ rows and $n$ columns. Each number in the matrix is called an entry or element, identified by its row and column position.

Think of a spreadsheet: rows go across, columns go down, and every cell holds a number. A $2 \times 3$ matrix is like a mini-spreadsheet with 2 rows and 3 columns. Matrices package multiple numbers into a single organized object so you can manipulate them all at once.

Read the full concept explanation →

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: A matrix is a rectangular array where each entry is pinned by its row and column address.

Common stuck point: The procedure for matrix definition is the easy part; the trap is stating dimensions columns-first. Asking "Are the numbers arranged in a fixed rectangle of rows and columns with a stated size?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the numbers arranged in a fixed rectangle of rows and columns with a stated size?

Worked Examples

Example 1

easy

Given

A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}

, what are the dimensions of

A

and what is

a_{2,3}

Answer

A

2 \times 3

;

a_{2,3} = 2

First step

Step 1: Count rows: 2. Count columns: 3. Dimensions:

2 \times 3

Full solution

2
Step 2: $a_{2,3}$ means row 2, column 3. That entry is $2$ .
3
Step 3: Verify: row 2 is $[0, 5, 2]$ , third element is $2$ ✓

A matrix's dimensions are always given as rows × columns. The notation

a_{i,j}

refers to the entry in row

i

and column

j

Example 2

medium

Write a

3 \times 1

column matrix where

a_{i,1} = 2i - 1

Example 3

challenge

Show how any

n\times n

matrix

A

decomposes uniquely as a sum

S + K

where

S

is symmetric and

K

is skew-symmetric.

Practice Problems

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

Example 1

easy

What are the dimensions of

\begin{bmatrix} 4 & 0 \\ -1 & 7 \\ 3 & 2 \end{bmatrix}

Example 2

medium

A

is a

4 \times 3

matrix, how many entries does it have? Can you multiply

A

by a

3 \times 5

matrix?

Example 3

easy

What are the dimensions of the matrix

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

Example 4

easy

In the matrix

A = \begin{pmatrix} 7 & 8 \\ 9 & 10 \end{pmatrix}

, what is the entry

a_{21}

Example 5

easy

How many entries does a

3 \times 4

matrix have?

Example 6

easy

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

a square matrix?

Example 7

easy

What is the

3 \times 3

identity matrix?

Example 8

easy

Give the dimensions of a column vector with 4 entries.

Example 9

easy

What is the entry

a_{12}

\begin{pmatrix} 5 & -3 \\ 2 & 8 \end{pmatrix}

Example 10

easy

A matrix has the same number of rows and columns. What is it called?

Example 11

medium

Matrix

B

2 \times 5

and matrix

C

5 \times 3

. What are the dimensions of

BC

Example 12

medium

Can a

3 \times 2

matrix be added to a

2 \times 3

matrix? Explain.

Example 13

medium

How many entries lie on the main diagonal of a

5 \times 5

matrix?

Example 14

medium

Matrix

A

4 \times 3

. For the product

AB

to be defined, what must be true of

B

's number of rows?

Example 15

medium

Two matrices are equal.

A = \begin{pmatrix} x & 2 \\ 3 & y \end{pmatrix}

and

B = \begin{pmatrix} 5 & 2 \\ 3 & 7 \end{pmatrix}

. Find

x

and

y

Example 16

medium

Is the product of a

1 \times 3

row vector and a

3 \times 1

column vector a scalar or a matrix?

Example 17

medium

Write the transpose of

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

Example 18

medium

A matrix is

m \times n

. After transposing twice, what are its dimensions?

Example 19

medium

4 \times 4

matrix has entry rule

a_{ij} = i + j

. Find

a_{23}

Example 20

challenge

A square matrix equals its own transpose. The entry

a_{13} = 6

. What must

a_{31}

be, and what is such a matrix called?

Example 21

challenge

A

m \times n

and

A = A^{T}

is possible, what relationship must

m

and

n

have, and why?

Example 22

challenge

A matrix is

n \times n

with all entries 0 except

a_{ii} = i

. Find the entry in row 3, column 3 and the total of all diagonal entries for

n = 4

Example 23

easy

How many entries does a

6 \times 7

matrix have?

Example 24

easy

True or false: a

1 \times 1

matrix is just a scalar.

Example 25

easy

What is the dimension of a row vector with 6 entries?

Example 26

easy

Is a matrix with all entries equal to

1

called the identity matrix?

Example 27

medium

Write a

3 \times 2

matrix

A

with

a_{ij}=i+2j

Example 28

medium

For two matrices to be equal, what conditions must hold?

Example 29

medium

Matrix

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

. Find

a

and

b

Example 30

medium

What is the transpose of

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

Example 31

medium

A matrix

A

7\times 4

. What are the dimensions of

A^T

Example 32

medium

A diagonal matrix has zeros everywhere except on the main diagonal. Write the

3 \times 3

diagonal matrix with diagonal

(2,5,-1)

Example 33

medium

How many distinct

2\times 2

matrices with entries in

\{0,1\}

are there?

Example 34

medium

For

A

to be square, what relationship must hold between its row count

m

and column count

n

Example 35

medium

A matrix is upper triangular if

a_{ij}=0

whenever

i>j

. Write a

3\times 3

upper triangular matrix with all diagonal entries

1

and superdiagonal entries

2

Example 36

medium

True/false: every

n\times n

symmetric matrix has

a_{ij}=a_{ji}

for all

i,j

Example 37

hard

How many distinct

3\times 3

diagonal matrices with diagonal entries in

\{0,1\}

are there, and how many of them are the identity or zero?

Example 38

hard

A skew-symmetric matrix satisfies

A^T=-A

. Why must its diagonal entries all be zero?

Example 39

hard

A matrix

A

3\times 4

. Can

A^T A

be defined, and if so what are its dimensions?

Example 40

hard

Write the

3\times 3

permutation matrix that sends the standard basis vector

e_1 \to e_2,\; e_2\to e_3,\;e_3\to e_1

Example 41

hard

For an

n\times n

symmetric matrix, how many independent entries are there?

Example 42

challenge

How many independent entries does an

n\times n

skew-symmetric matrix have?

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Related Concepts

Systems of Equations Variables Matrix Addition, Subtraction, and Scalar Multiplication Matrix Multiplication Determinant

Background Knowledge

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

systems of equationsvariables