Download Notes 3.8 Identity and Inverse Matrices Think back to chapter 1

Notes 3.8 Identity and Inverse Matrices
Think back to chapter 1 when we went over some of the properties. What did the Identity Property of Multiplication tell
us? ______________________________________________________________________________________________
What did the Inverse Property of Multiplication tell us? ____________________________________________________
If I gave you the matrix A = 3 -2
-1 4
could you come up with a 2 x 2 matrix that, when multiplied with matrix A, gives you back the same matrix? (Hint: what
numbers would we want to try that would either keep numbers the same or not cause anything to be added or
subtracted?)
Show what you tried below and what you found worked:
The multiplicative identity for 2 x 2 matrices is:
The multiplicative identity for 3 x 3 matrices is:
How do I know something is an inverse of another? ________________________________________________________
Two matrices are inverses of each other if _______________________________________________________________
The inverse of a 2 x 2 matrix A = a b
c d
is A -1 =
Ex 1: Find the inverse of A = 2 4
1 3
Ex 2: Solve the matrix equation
11 5 X = 4 -1
2 1
2 0
Some matrices do not have an inverse. What do you think would cause this? (Hint: it has to do with the determinant.)
_________________________________________________________________________________________________
To verify an inverse, you must be able to multiply it both ways and get the identity inverse. Finding the inverse of a 3 x 3
matrix is much more difficult to do by hand than a 2 x 2, but is easy when utilizing a graphing calculator. We will cover
this another day. However, if we are given a matrix and a possible inverse, we can still verify it through matrix
multiplication.
Ex 3: Verify that B = 1 -1 0
1 0 -1
6 -2 -3
and
A = -2 -3 1
-3 -3 1
-2 -4 1
are inverses of each other.