Matrices

A matrix is a rectangular array of numbers enclosed by brackets. Matrices (the plural form of a matrix) have a lot of uses in math, including solving systems of linear equations and solving algebra problems. Matrices are also used in advanced mathematics. To give you an introduction to the matrix, let''s go over its basics.

An overview of matrices: dimensions

When looking at a matrix, you''ll notice that it consists of one or more numbers that line up in rows (horizontal) and columns (vertical). Each number in a matrix is called an element (or entry) of the matrix.

The number of rows and columns determines the matrix dimensions. When writing the dimensions, you''ll always write the number of rows first and the number of columns second.

Let''s look at examples:

$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 7& -1\end{array}\right]$

The above matrix consists of three columns and two rows. Its dimensions are $3\times 2$ (or 3 by 2).

$\left[\begin{array}{ccc}6& -2& -1\end{array}\right]$

This matrix consists of one row and three columns, so its dimensions are $1\times 3$ .

$\left[\begin{array}{c}-5\\ 3\\ 10\end{array}\right]$

This matrix consists of three rows and one column, so its dimensions are $3\times 1$ .

$\left[\begin{array}{cc}1& 3\\ -1& -9\end{array}\right]$

This final matrix consists of two rows and two columns, so its dimensions are $2\times 2$ .

Types of matrices

Understanding the basics of matrix dimensions will make it easier to identify the different types of matrices you''ll encounter. Here are a few matrix types:

• Row matrix: A row matrix has only one row and any number of columns. The following is a $1\times 4$ matrix.

  $\left[\begin{array}{cccc}5& -4& 3& 1\end{array}\right]$

• Column matrix: A column matrix has one column and any number of rows. The following is a $5\times 1$ matrix.

  $\left[\begin{array}{c}2\\ -4\\ 1\\ 5\\ 2\end{array}\right]$

• Square matrix: A square matrix always has the same number of rows as columns. The following is a $2\times 2$ square matrix.

  $\left[\begin{array}{cc}7& -1\\ 6& 5\end{array}\right]$

  And this is a $3\times 3$ square matrix:

  $\left[\begin{array}{ccc}3& -4& 6\\ 9& 4& 3\\ 2& -5& 8\end{array}\right]$

• Zero matrix: A matrix with all zero elements is called a zero matrix. The following is a $2\times 3$ zero matrix, denoted $0_{2\times 3}$

  $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$

• Rectangular matrix: A rectangular matrix means that the number of columns is not equal to the number of rows.

  $\left[\begin{array}{cc}6& 4\\ 2& 7\\ 3& -9\end{array}\right]$

Adding and subtracting matrices

It''s only possible for a matrix to add to or subtract from another matrix if the two matrices have the same dimensions. Let''s look at some examples.

Adding matrices

In order to add two matrices, you''ll simply add the corresponding entries, and then place the sum in the corresponding position in the resulting matrix.

Let''s add the following matrices:

$\left[\begin{array}{cc}8& -1\\ -4& -3\end{array}\right]+\left[\begin{array}{cc}-3& 9\\ 5& 2\end{array}\right]$

Before getting started, it''s always important to make sure the two matrices you''re adding have the same dimensions. In this case, both are $2\times 2$ square matrices, so we''re all set to add the entries.

$\left[\begin{array}{cc}8& -1\\ -4& -3\end{array}\right]+\left[\begin{array}{cc}-3& 9\\ 5& 2\end{array}\right]=\left[\begin{array}{cc}8+\left(-3\right)& -1+9\\ -4+5& 3+2\end{array}\right]$

Than resulting matrix is:

$\left[\begin{array}{cc}5& 8\\ 1& 5\end{array}\right]$

Subtracting matrices

When subtracting matrices, you''ll subtract entries in the second matrix from the corresponding entries in the first matrix. Then, you''ll place the sum in the corresponding position in the resulting matrix.

$\left[\begin{array}{cc}8& 7\\ -4& 6\end{array}\right]-\left[\begin{array}{cc}5& 3\\ 1& 2\end{array}\right]$

Again, we want to start by making sure that the two matrices have the same dimensions. Since they''re both $2\times 2$ matrices, we''re all set to begin.

$\left[\begin{array}{cc}8& 7\\ -4& 6\end{array}\right]-\left[\begin{array}{cc}5& 3\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}8-5& -4-1\\ 7-3& 6-2\end{array}\right]$

Than resulting matrix is:

$\left[\begin{array}{cc}3& 4\\ -5& 4\end{array}\right]$

Practice questions on matrices

1. What are the dimensions of the following matrix?

   $\left[\begin{array}{ccccc}2& -6& -7& 9& -1\end{array}\right]$

   Answer: $1\times 5$

2. What are the dimensions of the following matrix?
   

   $\left[\begin{array}{ccc}2& -3& 5\\ 7& 2& 1\\ 6& 8& -2\\ 4& 9& 6\end{array}\right]$

   Answer: $4\times 3$

3. What type of matrix is this?

   $\left[\begin{array}{c}-2\\ 6\\ 3\\ 4\end{array}\right]$

   Answer: Column matrix

4. What type of matrix is this?
   

   $\left[\begin{array}{ccc}1& 5& 6\\ 4& 3& 7\\ 9& 2& 5\end{array}\right]$

   Answer: Square matrix

5. Add the following matrices:

   $\left[\begin{array}{cc}-3& 7\\ 6& 5\end{array}\right]+\left[\begin{array}{cc}8& 1\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}-3+8& 7+1\\ 6+2& 5+4\end{array}\right]$

   Answer: $\left[\begin{array}{cc}5& 8\\ 8& 9\end{array}\right]$

6. Subtract the following matrices:

   $\left[\begin{array}{cc}2& 4\\ 8& 7\end{array}\right]-\left[\begin{array}{cc}1& 2\\ 9& 3\end{array}\right]=\left[\begin{array}{cc}2-1& 4-2\\ 8-9& 7-3\end{array}\right]$

   Answer: $\left[\begin{array}{cc}1& 2\\ -1& 4\end{array}\right]$

Topics related to the Matrices

Multiplying Vector by a Matrix

Transformation of Graphs using Matrices - Translation

Adjoint of a Matrix

Flashcards covering the Matrices

Precalculus Flashcards

CLEP Precalculus Flashcards

Practice tests covering the Matrices

Precalculus Diagnostic Tests

Get a better understanding of matrices

Matrices are unique in appearance, which can make them a bit intimidating for first-time learners. If your student is having a hard time grasping how to write out matrix dimensions, would like a better understanding of different types of matrices, or wants to know more about adding and subtracting matrices, tutoring is a great idea. An experienced tutor can clarify confusing topics and provide valuable assistance with homework and test preparations. Reach out to the Educational Directors at Varsity Tutors to learn more about the perks of tutoring today.