Inverses

Review of Inverses

There are many types of inverses, additive inverse, multiplicative inverses, inverse of square roots, inverse of trig functions.  In each case, two inverses combine to give the identity element of that process.

The identity element of addition is 0.  So adding two inverses will result in 0.

3 + (-3) = 0

-10 + 10 = 0

The identity element of multiplication is 1.  So multiplying two inverses will result in 1.

3(1/3) = 1

-8(-1/8) = 1

Identity Matrix Under Multiplication

The following are requirements for a matrix to be an Identity Matrix under multiplication:

1. It is a square matrix.
2. It has 1s on the diagonal and 0s everywhere else.
3. It it denoted by the capital letter I.

Multiplicative ID Matrices

Multiplicative Inverse of a Matrix

Inverse Matrix: The inverse of A is called A^-1 iff AA^-1 = A^-1A = I
Singular: A matrix that does not have an inverse.

Example: Show that A and B are inverses.

Solution

Since the product is the identity matrix under multiplication, A and B are inverses.


Are They Inverses? Practice: Given the matrices, answer the following questions.

#1)  Are A and B inverses?
#2)  Are C and D inverses?

Multiplicative Inverse of a 2nd-Order Matrix

Definition

Given

Then 

determinant reciprocal times matrix with blues swapped and reds' signs changed

Example: Find the inverse of the matrix

blues were swapped and reds' signs were changed

Multiplicative Inverse Practice: Find each inverse if possible.  If not possible, write singular.

#1) A^-1
#2) B^-1

Homework

Are They Inverses?

Determine if each pair of matrices are inverses.

#1) Are A and B inverses?







#2) Are C and D inverses?






#3) Are E and F inverses?






#4) Are G and H inverses?







#5) Are J and K inverses?







#6) Are L and M inverses?

Multiplicative Inverses

Find the multiplicative inverse of each matrix. If not possible, write singular.

#1) A^-1
#2) B^-1
#3) C^-1
#4) D^-1
#5) E^-1
#6) F^-1

Answers Are They Inverses?

#1) No
#2) No
#3) Yes
#4) No
#5) Yes
#6) No

Answers Multiplicative Inverses

2) B is singular