How to Find the Inverse of a Matrix: Special Cases
In this lesson, we show how to find the
inverse
of a
matrix for two
special cases: a
diagonal matrix
and a 2 x 2 matrix. In the next lesson, we show
how to find the inverse for any matrix.
How to Find the Inverse of a Diagonal Matrix
A diagonal matrix matrix is a special kind
of
symmetric matrix.
It is a symmetric matrix with zeros in the
offdiagonal elements. Two diagonal matrices are shown below.
Note that the diagonal of a matrix refers to the elements
that run from the upper left corner to the lower right corner.
The inverse of a diagonal matrix is obtained by replacing each
element in the diagonal with its reciprocal, as illustrated
below for matrix C.
It is easy to confirm that C^{1} is the
inverse of C, since
CC^{1}
= C^{1}C
= I
where I is the
identity matrix.
This approach will work for any diagonal matrix, as long as none of the
diagonal elements is equal to zero. If any of the diagonal elements
are equal to zero, the matrix will be less than
full rank,
and the matrix will not have an inverse.
How to Find the Inverse of a 2 x 2 Matrix
Suppose A is a
nonsingular matrix
2 x 2 matrix. Then, the inverse of A can be computed
from A, as shown below.

A_{1}_{1} 
A_{1}_{2} 

A_{2}_{1} 
A_{2}_{2} 



A_{2}_{2}/A 
A_{1}_{2}/A 

A_{2}_{1}/A 
A_{1}_{1}/A 

A 

A^{1} 
where the
determinant
of A is
A = A_{1}_{1}A_{2}_{2}  A_{1}_{2}A_{2}_{1} .
To illustrate how this works, let's find the inverse of matrix
B, which appears below.
First, let's compute the determinant of matrix B.
B = B_{1}_{1}B_{2}_{2}
 B_{1}_{2}B_{2}_{1} = 2*4  1*4 = 8  4 = 4
Then, we can find the inverse, as shown below.
B^{1} = 

B_{2}_{2}/B 
B_{1}_{2}/B 

B_{2}_{1}/B 
B_{1}_{1}/B 

Warning: If the determinant of a matrix is
equal to zero, then the matrix does not have an inverse.
Test Your Understanding
Problem 1
Find the inverse of matrix A, shown below.
Solution
This was sort of a trick question.
Matrix A is a diagonal matrix with a zero element in
its diagonal. Therefore, matrix A is singular, and
does not have an inverse.
Problem 2
Find the inverse of matrix A, shown below.
Solution
The inverse of a diagonal matrix is obtained by replacing each element
in the diagonal with its reciprocal, as shown below.
Problem 3
Find the inverse of matrix A, shown below.
Solution
First, let's compute the determinant of matrix A.
A = A_{1}_{1}A_{2}_{2}
 A_{1}_{2}A_{2}_{1} = 3*4  1*9 = 12  9 = 3
Then, we can find the inverse, as shown below.
A^{1} = 

A_{2}_{2}/A 
A_{1}_{2}/A 

A_{2}_{1}/A 
A_{1}_{1}/A 
