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×nm \times n matrix has mm rows and nn 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 2×32 \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

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Two matrices are equal. A=(x23y)A = \begin{pmatrix} x & 2 \\ 3 & y \end{pmatrix} and B=(5237)B = \begin{pmatrix} 5 & 2 \\ 3 & 7 \end{pmatrix}. Find xx and yy.

Example 2

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A matrix AA is 7×47\times 4. What are the dimensions of ATA^T?

Example 3

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A matrix is m×nm \times n. After transposing twice, what are its dimensions?

Example 4

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

Example 5

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A 4×44 \times 4 matrix has entry rule aij=i+ja_{ij} = i + j. Find a23a_{23}.

Example 6

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What are the dimensions of [40−1732]\begin{bmatrix} 4 & 0 \\ -1 & 7 \\ 3 & 2 \end{bmatrix}?

Example 7

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How many independent entries does an n×nn\times n skew-symmetric matrix have?

Example 8

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How many entries does a 3×43 \times 4 matrix have?

Example 9

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Identify the main diagonal entries of A=(314159265)A=\begin{pmatrix} 3 & 1 & 4 \\ 1 & 5 & 9 \\ 2 & 6 & 5 \end{pmatrix}.

Example 10

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Can a 3×23 \times 2 matrix be added to a 2×32 \times 3 matrix? Explain.

Example 11

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True or false: a 1×11 \times 1 matrix is just a scalar.

Example 12

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What is the entry a12a_{12} in (5−328)\begin{pmatrix} 5 & -3 \\ 2 & 8 \end{pmatrix}?

Example 13

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For A=(123456789)A=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, what is a32a_{32}?

Example 14

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

Example 15

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What are the dimensions of (10000100)\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}?

Example 16

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True/false: every n×nn\times n symmetric matrix has aij=ajia_{ij}=a_{ji} for all i,ji,j.

Example 17

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Matrix A=(1a23)=(b423)=BA=\begin{pmatrix} 1 & a \\ 2 & 3 \end{pmatrix}=\begin{pmatrix} b & 4 \\ 2 & 3 \end{pmatrix}=B. Find aa and bb.

Example 18

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

Example 19

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What is the transpose of (123456)\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}?

Example 20

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Write a 3×13 \times 1 column matrix where ai,1=2i−1a_{i,1} = 2i - 1.
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