Introduction to Matrices

Section 5.1 - Introduction to Matrices

We recall that a matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has
size (order) m x n. The entry in the ith row and jth column is denoted by a_ij.

Example 1: Suppose A is a matrix of order 3 x 2 and a_ij = i-3j for all i and j. Find the matrix A.

Definitions:

1. A row matrix is a matrix of size 1 x n.
2. A column matrix is a matrix of size m x 1.
3. A matrix is called a MATH -software/decimal-to-square-feet-convert.html">square matrix if the number of rows equals the number of columns.
4. Two matrices are equal if they have the same size (order) and equal corresponding entries.

Example 2: Given the fol lowing matrix eqation, what are the values of a, b, and c?

Matrix Ope rations :
1. Addition (You can only add matrices that have the same size)
Example 3: Perform the following matrix operation:

2. Subtraction (You can only subtract matrices that have the same size)
Example 4: Perform the following matrix operation:

3. Transpose: The transpose of an m x n matrix A with entries a_ij is the n m matrix A^T with entries a_ji.
Example 5: Find A^T if

4. Scalar Product: For a matrix A and a real number c, the scalar product cA is found by multiplying each
entry in A by the real number c.
Example 6: Find 3A if

We can allow the calculator to do these basic calculations for us.

(If you have a plain TI-83 (without the Plus), when you see directions to hit 2nd x^-1, you need to hit the MATRIX button.)

Enter the matrix into the calculator:

 •Hit 2nd x^-1.
 •Cursor right two places to EDIT and hit ENTER on the matrix you wish to edit.
 •Enter the size of the matrix.
 •Enter the matrix elements.

Call a matrix for a computation:

 •Make sure you are on the home screen before you begin any calculations. To do this, hit 2nd MODE to quit.
 • Press 2nd, x^-1 and cursor down under the x^-1 list until you get to the matrix you want and hit
ENTER. The name of the matrix you need to do computations with will now be on your homescreen.

Take the Transpose of a Matrix

 •Call the matrix you would like to transpose from the home screen.
 •Press 2nd, x^-1, cursor right to MATH and select option 2:T.
 •You should now see the symbolic representation of transposing your matrix. To actually see the transpose,
hit ENTER.

Example 7: Given the following matrices, perform the following operations on your calculator .

a) Find A+C

b) Find A-B

c) Find C^T

Section 5.2 - Matrix Multiplication

How to Multiply Matrices (C = AB):

1. Check to see if the number of columns of matrix A is equal to the number of rows in matrix B. If this
condition is satisfied, the multipication is a valid operation. The resulting matrix C will have the same
number of rows as A and the same number of columns as B.

2. Compute each entry in the resulting matrix C. The entry c_ij is found by using the ith row of matrix A
and the jth column of matrix B as shown in the next example.

Example 1: Let

Find C where C = AB.

Note: In general, AB≠BA

Example 2: Given the following matrices with given dimensions, de termine whether each of the following
is a valid matrix operation.

Definition: The identity matrix of size n has n rows and n columns. It has 1’s along the main diagonal and
0’s everywhere else.

A matrix multiplied by the appropriate identity matrix results in the orignal matrix.

Example 3: Mulitply the matrix below by the appropriate identiy matrix.

Example 4: Find a, b, and c in the following matrix equation .

Example 5: The Yummy Cake Company makes three types of cakes: Angel Food, Italian Cream, and
Chocolate. The company produces its cakes in College Station, Santa Cruz, and Austin using two main
ingredients, sugar and flour.

a) Each ki logram of Angel Food requires 0.1 kg of sugar and 0.5 kg of flour. Each kilogram of Italian
Cream requires 06 kg of sugar and 0.2 kg of flour. Each kilogram of Chocolate requires 0.3 kg of
sugar and 0.3 kg of flour. Put this information into a 2 x 3 matrix.

b) The cost of 1 kg of sugar is $1 in College Station, $4 in Santa Cruz and $2 in Austin. The cost of 1
kg of flour is $0.50 in College Station, $1.50 in Santa Cruz and $1 in Austin. Put this information
into a matrix in such a way that when it is multiplied by the matrix in part a) it will tell us the cost
of producing each kind of cake in each city. Find the resulting product matrix.