Fundamentals of matrices 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 2 3 x 1 1 x 3

EXAMPLE 1 Add and subtract matrices Perform the indicated operation, if possible. 3 0 –5 –1 a. –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. – 7 – (–2) 4 – 5 0 – 3 –2 – (–10) –1 – (–3) 6 – 1 = 9 –1 –3 8 2 5 =

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) = –8 2 –2 0 –4 –14 = b. 4 –2 –8 5 0 –3 8 6 –5 + 4(–2) 4(–8) 4(5) 4(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. a. A: 4 x 3, B: 3 x 2 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.

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

EXAMPLE 3 Find the product of two matrices Find AB if A = 1 4 3 –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 4 3 –2 5 –7 9 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 4 3 –2 5 –7 9 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 4 3 –2 5 –7 9 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 4 3 –2 5 –7 9 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 =

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”.