ch 3.5-3.6 fundamentals of matrices matrix - a rectangular arrangement of numbers in rows and...
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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”.