Warm-Up 3) 1) 4) Name the dimensions 2)

Multiplying Matrices

Can You Multiply Matrices? What do you think has to be true in order to multiply? What procedure do you think we’d use? Let’s try this one…  

When Can You Multiply Matrices? Was your answer… ? Then you are WRONG! The correct answer is…    

So, How Do We Multiply Matrices? What ideas did you come up with regarding WHEN we CAN multiply matrices? Two matrices can be multiplied if and only if the number of ______________ in the first one is EQUAL to the number of ______________ in the second one.

In order to multiply matrices... A • B = AB m x n n x p m x p Ex 1. Can you multiply? What will the dimensions be? A B AB 2 x 3 3 x 4 2 x 4 AB A B 5 x 3 5 x 2 Not possible

So, now, how do we find the entries? The reason the columns and rows must match up is the process used to determine the entries... I call this process “The Finger Method” Looking back at our original problem…  

Next, what will the dimensions of the answer be?   Next, what will the dimensions of the answer be? First, check the dimensions… 2 x 2 2 x 2 The dimensions of the answer will be 2x2. They match… so we can multiply! Now draw a BLANK matrix with the correct dimensions for the answer…  

  To find the entry for the FIRST ROW , 2nd COLUMN use the FIRST ROW and 2nd COLUMN (Remember to always start at the beginning of the row and top of the column) To find the entry for the FIRST ROW , FIRST COLUMN use the FIRST ROW and FIRST COLUMN (Remember to always start at the beginning of the row and top of the column) To find the entry for the 2nd ROW , 1st COLUMN use the 2nd ROW and 1st COLUMN (Remember to always start at the beginning of the row and top of the column) To find the entry for the 2nd ROW , 2nd COLUMN use the 2nd ROW and 2nd COLUMN (Remember to always start at the beginning of the row and top of the column)   1(2) + 2(-1) 1(3) + 2(5) 4(2) + 2(-1) 4(3) + 2(5)

    1(2) + 2(-1) 1(3) + 2(5) 4(2) 4(3) =   = 13 6 22

The Associative Property The Associative Property for Multiplication states (A x B) x C= A x (B x C) For Matrices, this is also true… This means that you can multiply either set first and get the same answer…

Given that we just found…           What do you think equals?

The Commutative Property The Commutative Property for Multiplication states A x B = B x A But, in this last example we found that it does not hold true. In fact, it is rare for matrices to be commutative with respect to multiplication…

The Commutative Property Since the Commutative Property for Multiplication does not hold true for Matrices, it is extremely important that you remember… NEVER SWITCH THE ORDER OF MATRICES WHEN MULTIPLYING!!!

How to multiply... ac ad = 2 x 1 1 x 2 2 x 2

How to multiply... ac ad = bc bd 2 x 1 1 x 2 2 x 2

Ex. 1

Ex. 2 Find AB -16 + 1 -12 +2 -15 -10 = - 2 - 4 -2 - 4

Ex. 3 Find BA

Ex. 4 -1(4) +5(6) -1(-3) +5(8) -4 + 30 3 + 40 5(4) +2(6) 5(-3) +2(8) 20 +12 -15 + 16 = + -24 + -32 0(4) +-4(6) 0(-3) +-4(8) 26 43 32 1 -24 -32

What are your questions?

Assignment