Topics
14. Matrices and Networks
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Middle Years

14.02 Matrix addition, subtraction and scalar multiplication

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Worksheet
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Lesson

When we add or subtract two matrices $A$A and $B$B, we add or subtract the corresponding elements. So we can only add or subtract matrices that have the same dimensions. In general, the sum and difference of two matrices can be represented similar to the following:

Addition Subtraction

 

As an example, the sum and difference of two matrices with numerical elements will look like the following:

Addition Subtraction

 

A scalar is a quantity, or a magnitude. When we multiply a matrix by a scalar, we multiply each element in the matrix by the given scalar. In general, the multiplication of a matrix by a scalar will appear similar the following:

Multiplication by a scalar

 

As an example, the multiplication of a matrix with numerical elements by a scalar will look like the following:

 

Worked examples

example 1

Evaluate the following multiplication of a matrix by scalar.

Think: When we want to evaluate the scalar multiplication of a matrix, we multiply each element by the scalar.

Do: Multiplying each element by $\frac{1}{2}$12​ is the same as dividing each element by $2$2. This gives us the following matrix.

example 2

Rewrite the following as a single matrix.

Think: The order of operations of matrices follow the typical order of operations of numbers. So we first want to compute the scalar multiplication and then compute the subtraction.

Do: Notice that we've already solved the scalar multiplication in the previous example. So we can simplify the matrix expression as the following.

To compute the difference between the matrices, we subtract the corresponding elements which gives us the following.

Practice questions

Question 1

Consider the matrices

$A$A$=$= and $B$B$=$=

  1. Matrix $A$A has dimensions $\editable{}$$\times$×$\editable{}$

  2. Matrix $B$B has dimensions $\editable{}$$\times$×$\editable{}$

  3. Is $A+B$A+B possible?

    Yes

    A

    No

    B

question 2

    $7$7         $8$8    
If $A$A$=$=   $-7$−7     and $B$B$=$=   $2$2   , find $A+B$A+B.
    $-5$−5         $4$4    
  1.     $\editable{}$    
    $A+B$A+B$=$=   $\editable{}$    
        $\editable{}$    

question 3

If $A$A$=$=   $9$9 $2$2 $6$6     and $B$B$=$=   $-3$−3 $-5$−5 $3$3   , find $A-B$A−B.
  $5$5 $-7$−7 $8$8       $-1$−1 $-6$−6 $7$7  
  1. $A-B$A−B$=$=   $\editable{}$ $\editable{}$ $\editable{}$    
      $\editable{}$ $\editable{}$ $\editable{}$    

question 4

If $A$A$=$=
    $2$2 $-6$−6    
    $-4$−4 $3$3    
, find $3A$3A.
  1. $3A$3A $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    

question 5

Find $A+5B$A+5B if $A$A$=$=
    $6$6 $-3$−3    
    $9$9 $7$7    
and $B$B$=$=
    $5$5 $0$0    
    $-4$−4 $2$2    
.
  1. $A+5B$A+5B$=$=
        $6$6 $-3$−3    
        $9$9 $7$7    
    		$+$+ $\editable{}$
        $5$5 $0$0    
        $-4$−4 $2$2    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    		$+$+
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    

 

Matrices and using the calculator

Matrices allow us to perform numerous calculations in one operation. They are very useful when analysing and keeping track of large amounts of data. For example, we could store information about a large inventory in a matrix. We could then use spreadsheets, calculators or other technology to monitor stock, by using matrix addition to add in orders, subtraction to take away sales and multiplications to calculate revenue, expenses and profit. 

Let's look at how we can enter matrices and use matrix operations with our calculator.

TI Nspire

How to use the TI Nspire to complete the following tasks regarding matrices

If $A$A$=$=    $1$1  $5$5  $6$6     and $B$B$=$=    $4$4  $1$1  $0$0    
   $3$3  $4$4  $5$5       $-2$−2  $5$5  $1$1  

  1. Find $3A$3A.

  2. Find $A+2B$A+2B.

  3. Find $3B-2A$3B−2A.

 

Try using your calculator to find the answers to the following practice questions, click reveal solution to check your final answer.

Practice questions

Question 6

Find $\frac{1}{2}A+\frac{2}{3}B$12​A+23​B if $A$A$=$=
    $6$6 $-12$−12    
    $4$4 $14$14    
and $B$B$=$=
    $24$24 $3$3    
    $-12$−12 $0$0    
.
  1. $\frac{1}{2}A+\frac{2}{3}B=$12​A+23​B= $\editable{}$
        $6$6 $-12$−12    
        $4$4 $14$14    
    		$+$+ $\editable{}$
        $24$24 $3$3    
        $-12$−12 $0$0    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    		$+$+
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    

Question 7

Let $A$A$=$=
    $10$10 $4$4    
    $3$3 $-9$−9    
, $B$B$=$=
    $1$1 $6$6    
    $7$7 $-1$−1    
and $C$C$=$=
    $0$0 $-2$−2    
    $-5$−5 $8$8    
.

  1. Find $4A+2B-5C$4A+2B−5C.

    $4A+2B-5C$4A+2B−5C$=$= $\editable{}$
        $10$10 $4$4    
        $3$3 $-9$−9    
    		$+$+ $\editable{}$
        $1$1 $6$6    
        $7$7 $-1$−1    
    		$-$− $\editable{}$
        $0$0 $-2$−2    
        $-5$−5 $8$8    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    		$+$+
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    		$-$−
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
    $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    

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