1. Fundamentals of matricesCh
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 2
3 x 1
1 x 3

2. EXAMPLE 1
Add and subtract matrices
Perform the indicated operation, if possible.
–5 –1 a.
–1 4
2 0 +
3 + (–1) – –1 + 0 = =
–3 –1 –
3 –10 –
0 –2
–1 6 b. –
7 – (–2) – 5
0 – –2 – (–10)
–1 – (–3) – 1 =
9 –1 – =

3. EXAMPLE 2
Multiply a matrix by a scalar
Perform the indicated operation, if possible. a.
4 –1
1 0
2 7 –2
–2(4) –2(–1)
–2(1) –2(0)
–2(2) –2(7) = – –
–4 –14 =
b. 4
–2 –8 – –5 +
4(–2) 4(–8) (5) (0)
–3 8
6 –5 = +
–8 –32
–3 8
6 –5 = +
–8 + (–3) –32 + 8
(–5) =
–8 + (–3) –32 + 8
(–5) =
–11 –24 –5 =

4. EXAMPLE 3
Describe matrix products
State whether the product AB is defined. If so, give the dimensions of AB.
a. A: 4 x 3, B: 3 x b. A: 3 x 4, B: 3 x 2
SOLUTION
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.
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.

5. 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
2. A: 3 x 2, B: 3 x 2
ANSWER
ANSWER
defined; 5 x 2
not defined

6. EXAMPLE 3
Find the product of two matrices
Find AB if A =
3 –2
and B = –7
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.

7. 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.
3 –2
5 –7
1(5) + 4(9) =

8. 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.
3 –2
5 –7
1(5) + 4(9) =
1( –7) + 4(6)

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

10. 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.
3 –2
5 –7 =
1(5) + 4(9)
1(–7) + 4(6)
3(5) + (–2)(9)
3(–7) + (–2)(6)

11. EXAMPLE 3
Find the product of two matrices
STEP 5
1(5) + 4(9) (–7) + 4(6)
3(5) + (–2)(9) (–7) + (–2)(6)
–3 –33 =

12. HERE HAS TO BE AN EASIER WAY!!!!!!!
EXAMPLE 3
HERE HAS TO BE AN EASIER WAY!!!!!!!
Why, yes there is. Please get out your calculator and your
Matrices worksheet entitled “Matrix Operations”.