4-5: Inverse of a square matrix
Objectives:
Finding inverse of a square matrix by reducing
matrix method.
Defn: The identity matrix for multiplication for the
set of all square matrices, of order n, is the square
matrix of order n, denoted by I, with 1’s along the
principal diagonal and 0’s elsewhere.
Ex:
Defn: Let M be a square matrix of order n and I be
the identity matrix of order n. If there exists a matrix
M-1 such that MM −1 = M −1M = I , then M-1 is called
the multiplicative inverse of M, or, more simply, the
inverse of M. If no such matrix exists, then M is said
to be a singular matrix.
Main idea:
How to find the inverse of a matrix
If [M|I] (the combined matrix of M and I) can be
reduced to [I|B] by row operations, then M-1=B. If
we obtain all 0’s in one or more rows to the left of
the vertical line, then M-1 does not exist.
Why?
 4 − 1
Ex. Find M , given M = 
.

− 6 2 
-1
Check:
 2 − 4
Ex. Find M if it exists, given M = 
.

− 3 6 
-1
1 1 1 
Ex. Find inverse of 2 1 2 .


2 3 1 
A shortcut to find inverse for a 2×2 matrix (pg224
Explore & Discuss 3):
a b 
If A = 
, then

c d 
1  d − b
, where the determinant of
A =


ad − bc − c a 
A is defined by det( A) = ad − bc
−1
1  d − b
So A =
det( A) − c a 
−1
If det( A) = 0 , then the inverse does not exist.
 4 − 1
Ex: M = 

− 6 2 
Ex: Without performing any row operations, explain
why each of the matrices does not have an inverse.
0 − 1 
1) 2 − 2


1 − 3
0 1 
3) 

0
6


 0 0
2) 

 − 3 6
 2 3
4) 

4
6


4 0 0 
Ex. Find inverse of 0 6 0  mentally.


0 0 − 2