Ch 3.5-3.6 Fundamentals of matrices

Matrix - a rectangular arrangement of numbers in rows and columns.

- have the size or “dimensions” of m x n• m = # horizontal rows• n = # vertical columns

- “equal” matrices have the same dimensions and the same elements.

3 x 13 x 2 1 x 3

EXAMPLE 1 Add and subtract matrices

Perform the indicated operation, if possible. 3 0 –5 –1a.

–1 4 2 0+

3 + (–1) 0 + 4 –5 + 2 –1 + 0= =

2 4 –3 –1

–2 5 3 –10–3 1

7 4 0 –2 –1 6

b. –

9 –1–3 8 2 5

= 7 – (–2) 4 – 5 0 – 3 –2 – (–10) –1 – (–3) 6 – 1

=

EXAMPLE 2 Multiply a matrix by a scalar

Perform the indicated operation, if possible.

4(–2) 4(–8) 4(5) 4(0)

–3 8 6 –5

= +

a.4 –11 02 7

–2–2(4) –2(–1)–2(1) –2(0)–2(2) –2(7)

= –8 2 –2 0 –4 –14

=

b. 4–2 –8 5 0

–3 8 6 –5

+

–8 –32 20 0

–3 8 6 –5= +

–8 + (–3) –32 + 8 20 + 6 0 + (–5)

=

–8 + (–3) –32 + 8 20 + 6 0 + (–5)

= –11 –24 26 –5

=

EXAMPLE 3 Describe matrix products

State whether the product AB is defined. If so, give the dimensions of AB.

SOLUTION

b. Because the number of columns in A (four) does not equal the number of rows in B (three), the product AB is not defined.

a. A: 4 x 3, B: 3 x 2 b. A: 3 x 4, B: 3 x 2

a. Because A is a 4 x 3 matrix and B is a 3 x 2 matrix, the product AB is defined and is a 4 x 2 matrix.

GUIDED PRACTICE for Example 3

State whether the product AB is defined. If so, give the dimensions of AB.

1. A: 5 x 2, B: 2 x 2

defined; 5 x 2

ANSWER

not defined

2. A: 3 x 2, B: 3 x 2

ANSWER

EXAMPLE 3 Find the product of two matrices

Find AB if A =1 43 –2 and B = 5 –7

9 6

SOLUTION

Because A is a 2 X 2 matrix and B is a 2 X 2 matrix, the product AB is defined and is a 2 X 2 matrix.

EXAMPLE 3 Find the product of two matrices

STEP 1

Multiply the numbers in the first row of A by the numbers in the first column of B, add the products, and put the result in the first row, first column of AB.

1 43 –2

5 –79 6

1(5) + 4(9)=

EXAMPLE 3 Find the product of two matrices

STEP 2

Multiply the numbers in the first row of A by the numbers in the first column of B, add the products, and put the result in the first row, second column of AB.

1 43 –2

5 –79 6

1(5) + 4(9)=

1( –7) + 4(6)

EXAMPLE 3 Find the product of two matrices

STEP 3

Multiply the numbers in the second row of A by the numbers in the first column of B, add the products, and put the result in the second row, first column of AB.

1 43 –2

5 –79 6

1(5) + 4(9) 1(–7) + 4(6)3(5) + (–2)(9)=

EXAMPLE 3 Find the product of two matrices

STEP 4

Multiply the numbers in the second row of A by the numbers in the second column of B, add the products, and put the result in the second row, second column of AB.

1 43 –2

5 –79 6

=1(5) + 4(9) 1(–7) + 4(6)

3(5) + (–2)(9) 3(–7) + (–2)(6)

EXAMPLE 3 Find the product of two matrices

STEP 5

1(5) + 4(9) 1(–7) + 4(6)3(5) + (–2)(9) 3(–7) + (–2)(6)

41 17–3 –33

=

EXAMPLE 3

HERE HAS TO BE AN EASIER WAY!!!!!!!

Why, yes there is. Please get out your calculator and yourMatrices worksheet entitled “Matrix Operations”.