What is a Matrix?

A matrix is a rectangular array of numbers used to manage and organize data. It is made up of horizontal rows and vertical columns.



Dimensions of the Matrix (Am,n)
A matrix with m rows and n columns has dimensions of m ×n. For example, the matrix A2,3 has 2 rows and 3 columns.



Example 1

Types of Matrices

A matrix with only one row is called a row matrix. A matrix with only one column is called a column matrix. A matrix with equal dimensions (m=n) is called square matrix.

Identity Matrix

It is a matrix with 1's on the main diagonal (the diagonal which runs from the upper-left to the lower-right) and 0's everywhere else.

Transpose of the Matrix (At)

You can find the transpose of the matrix by interchanging the corresponding rows and columns.

Addition and Subtraction

Two or more matrices can be added or subtracted only if they have the same dimensions. To add or subtract matrices, add or subtract the corresponding elements of each matrix. In the example seen below, if you wanted to add Matrix A and Matrix B, you would add [a 1,1] with [b 1,1]. Since [a 1,1] is 20 and [b 1,1] is 11, you would simply add 20 to 11, giving you an answer of 31, which would be the [c 1,1] element of Matrix C.

Multiplication and Division

1. Multiplying Matrices

You can ONLY multiply the matrices if the inner dimensions are the same. For instance, Am,n and Bn,p. Since Matrix A is 2 by 2, and Matrix B is 2 by 2, you can multiply these two matrices because the column number for Matrix A = 2 and the row number for Matrix B = 2.

Example 1

Find Matrix A * Matrix B.

Solution

To find A * B, the matrices A, B must have the same inner-dimensions. When multiplying two matrices, the resulting matrix will have the dimensions of the row number of Matrix A and the column number of Matrix B. So a 2 by 2 matrix * a 2 by 2 matrix will result in a 2 by 2 matrix. Similarly, a 2 by 3 matrix * a 3 by 4 matrix will result in a 2 by 3 matrix.

c 1 by 1= first row in A * first column in B
=6*2+-5*7= -23
c 1 by 2 = first row in A * second column in B
= 6*9+-5*4= 34
c 2 by 1=3*2+0*7=6
c2*2=3*9+0*4=27

Example 2




Multiplying Matrices Video


2. Finding the inverse of a 2 by 2 matrix.

Example 1

Finding the inverse of a 2 by 2 matrix Video
(The video uses the formula shown above.)


Example 2



To find if any 2 by 2 matrix has an inverse or not.

It has an inverse when ad≠bc, and the flowchart shows that .



3. Multiplying matrices by a scalar

To multiply a matrix by a scalar, multiply each element by the scalar quantity. For the example below, you would multiply 5 (the scalar quantity) by each element in the matrix.


4. Dividing Matrices

Dividing Matrices isn't like multiplying. It's different in some ways. If you are to divide by a scalar, each element is divided by the scalar quantity. However, a matrix divided by another matrix is ALWAYS undefined.

Example 1

Simplify the following expression and find A:


Solution

To isolate for A we divide each element by 2. You should get this:



Example 2

Simplify the following expression:



Solution

Since you are trying to divide a matrix by another matrix, the answer is: UNDEFINED

Matrices with real world problems

As defined before, matrices are used to organize and store collections of data. So far we've only discussed how to solve matrix problems with textbook questions. However, this does not mean that matrices are not useful in the real world.

1. Solving real world matrix problems by hand

Example 1

Matrix A represents the proportion of students at a high school who have part-time jobs on Saturdays and the length of their shifts. Matrix B represents the number of students in each grade.





Solution

Since Matrix A and Matrix B have the same dimensions, you are able to multiply. The result will be a 3 by 2 matrix.







Around 73 males and 77 males work up to 4 hours, 140 males and 148 females work 4 to 6 hours, and 65 and 71 females work more than 6 hours on Saturdays.

2. Multiplying matrices in spreadsheets

Using a spreadsheet program, such as Microsoft Excel, it is possible to do matrix functions in them such as multiply.

Example 1

The following table shows the number and sex of full-time students enrolled each university in Ontario one year.



Solution

First, we fill in the spreadsheet in Microsoft Excel with the following information like this (values in the spreadsheet go up to T3 but the picture cuts it off):



There are two functions that we are going to use. We are going to use the MMULT function and the INDEX function. The problem with the MMULT function is that it only displays the first entry of the product matrix. We then use the INDEX function to look at a specific element in the product matrix. To use the MMULT function, click on the cell that you want the result to be displayed in. At the top, in the entry bar, type in the following: =MMULT(array1,array2)

array1 = the range of cells defining the first matrix
array2 = the range of cells defining the second matrix

To use the INDEX function, in the entry bar at the top, type in the following: =INDEX(MMULT(array1,array2),row,column)

row = the row you want the product matrix to display
column = the column you want the product matrix to display

The result should look like this:



3. Finding the inverses of matrices in Excel

Example 2

Using Excel, we will calculate the inverse of:



Solution

First, type the matrix into Excel. It should look like this:



Click on the cell D1. This is where the first element of the inverse will be. While that cell is clicked on, at the entry bar on the top, type in the following: =MINVERSE(A1:B2)

A1:B2 defines where the matrix is located on the spreadsheet. However, alike the MMULT function, the MINVERSE function only displays the first element of the matrix. In order to display the whole matrix at one time, click on the cell to the right of the one displaying the first element and type in the following: =INDEX(MINVERSE(A1:B2),1,2)

The ",1,2)" sets which element to be displayed. Since it says 1,2, the element of the first row and second column will be displayed. Continue this procedure until the whole matrix is shown. It should look like this:



Links
Definitons regarding matrices

References
Mathematics of Data Management , Grade 12, (MDM4U). McGraw-Hill Ryerson