13.5 PROPERTIES OF MATRIX

MULTIPLICATION

WARM-UP

Use the following matrices to find 1 and 2

and

1. AB 2. BA

A = 6 4−9 −6

⎡

⎣⎢

⎤

⎦⎥

B = −2 −43 6

⎡

⎣⎢

⎤

⎦⎥

What did you find about #1 and #2?

  So, we can say that matrix multiplication is

not .

The only matrix where order doesn’t matter when you multiply is called the

matrix.

they are not equal

commutative

identity

I2×2 = 1 0

0 1

⎡

⎣⎢

⎤

⎦⎥

LET , SIMPLIFY:

3.

4.

A =a1 b1

a2 b2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

I2×2 ⋅A

A⋅I2×2

What did you find when you multiplied matrix A by the identity matrix?

So, no matter which way you multiply the identity (left or right), you get the same answer!

you get matrix A both times

PROPERTIES OF MATRIX MULTIPLICATION FOR 2X2 MATRICES

Let A, B, and C be 2X2 matrices and a be a scalar: 1. Closure Property:  2. Associative Property:  3. Distributive Property:

  4. Identity Property:

5. Associative Property for Scalar:

6. Multiplicative Property of Zero Matrix:

AB ∈Sm×n

AB( )C =A BC( )

A B +C( ) =AB+ AC or B+C( )A=BA+CA

I ⋅A=A or A⋅I =A

a AB( ) = aA( )B=A aB( )

0 ⋅A=A⋅0 =0

USE THE FOLLOWING TO SHOW #5

5.

A = 6 4−9 −6

⎡

⎣⎢

⎤

⎦⎥

B = −2 −43 6

⎡

⎣⎢

⎤

⎦⎥

A + B( ) A−B( ) ≠A2 −B2

USE THE FOLLOWING TO SHOW #6

6.

A = 6 4−9 −6

⎡

⎣⎢

⎤

⎦⎥

B = −2 −43 6

⎡

⎣⎢

⎤

⎦⎥

A + B( ) A−B( ) =A2 −AB+ BA−B2