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 off-diagonal elements. Two
diagonal matrices are shown below.
5 0 0
1 0
A= B= 0 3 0
0 3
0 0 1
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.
2 0 1/2 0
C= C-1 =
0 4 0 1/4
It is easy to confirm that C-1 is the inverse of C, since
CC-1 = C-1C = 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.
A11 A12 A22/|A| -A12/|A|
A21 A22 -A21/|A| A11/|A|
A A-1
where the determinant of A is |A| = A11A22 - A12A21 .
To illustrate how this works, let's find the inverse of matrix B, which appears below.
2 1
B=
4 4
First, let's compute the determinant of matrix B.
|B| = B11B22 - B12B21 = 2*4 - 1*4 = 8 - 4 = 4
Then, we can find the inverse, as shown below.
B-1 = B22/|B| -B12/|B|
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 off-diagonal elements. Two
diagonal matrices are shown below.
5 0 0
1 0
A= B= 0 3 0
0 3
0 0 1
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.
2 0 1/2 0
C= C-1 =
0 4 0 1/4
It is easy to confirm that C-1 is the inverse of C, since
CC-1 = C-1C = 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.
A11 A12 A22/|A| -A12/|A|
A21 A22 -A21/|A| A11/|A|
A A-1
where the determinant of A is |A| = A11A22 - A12A21 .
To illustrate how this works, let's find the inverse of matrix B, which appears below.
2 1
B=
4 4
First, let's compute the determinant of matrix B.
|B| = B11B22 - B12B21 = 2*4 - 1*4 = 8 - 4 = 4
Then, we can find the inverse, as shown below.
B-1 = B22/|B| -B12/|B|
