Practice Matrix Definition in Math

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

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

medium

A matrix

A

7\times 4

. What are the dimensions of

A^T

Example 3

medium

A matrix is

m \times n

. After transposing twice, what are its dimensions?

Example 4

medium

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

Example 5

medium

4 \times 4

matrix has entry rule

a_{ij} = i + j

. Find

a_{23}

Example 6

easy

What are the dimensions of

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

Example 7

challenge

How many independent entries does an

n\times n

skew-symmetric matrix have?

Example 8

easy

How many entries does a

3 \times 4

matrix have?

Example 9

easy

Identify the main diagonal entries of

A=\begin{pmatrix} 3 & 1 & 4 \\ 1 & 5 & 9 \\ 2 & 6 & 5 \end{pmatrix}

Example 10

medium

Can a

3 \times 2

matrix be added to a

2 \times 3

matrix? Explain.

Example 11

easy

True or false: a

1 \times 1

matrix is just a scalar.

Example 12

easy

What is the entry

a_{12}

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

Example 13

easy

For

A=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

, what is

a_{32}

Example 14

easy

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

Example 15

easy

What are the dimensions of

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}

Example 16

medium

True/false: every

n\times n

symmetric matrix has

a_{ij}=a_{ji}

for all

i,j

Example 17

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 18

easy

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

Example 19

medium

What is the transpose of

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

Example 20

medium

Write a

3 \times 1

column matrix where

a_{i,1} = 2i - 1

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