The diagonal matrix has diagonal elements only and if the diagonals are 1 then the matrix is called and identity matrix . In this article, we discuss about diagonal matrix and properties. Then find the inverse of diagonal matrix.
What is a diagonal matrix ?
A diagonal matrix is a matrix whose diagonal entries are non-zero and all other entries are zero. For example,

The matrix above is a diagonal matrix whose diagonal entries are
and other entries are
.
Some Interesting Properties of Diagonal Matrix
The diagonal matrix have some interesting properties explain in this section with examples.
Addition Or Multiplication of Diagonal Matrices
Addition or multiplication of two or more diagonal matrices of same order will give a diagonal matrix of same order.
Example #1
Let
and
be two diagonal matrices of order
.
Therefore, adding
and
will give

Example #2
Let
and
be two diagonal matrices of order
.
Therefore, multiplying matrix
and
will give

Multiplication Of Diagonal Matrix With Other Matrices
Let be matrix of order
and
be a matrix of order
.
Example #3

Each element of diagonal matrix is multiplied with corresponding row elements of matrix
. For example, element
from diagonal matrix
is multiplied with all elements of first row in matrix
.
Example #4
Another case of multiplication is when matrix of order
is multiplied with a diagonal matrix
of order
.
Let
be a matrix of order
and
be a diagonal matrix of order

When matrix C$ is multiplied with diagonal matrix then each element of
is multiplied with corresponding columns in matrix
. For example, element
is multiplied with first column of the matrix
.
Commutative Property of Multiplication in Between Diagonal Matrices
If two matrices and
are diagonal matrices of same order
, then the multiplication is commutative.
Example #5
Let
be a diagonal matrix of order
and
be a diagonal matrix of order
Similarly,

Therefore, for diagonal matrices is true.
Diagonal Matrix is Symmetric
If is a diagonal matrix of order
then it is symmetric.

Example #6
Let
be a diagonal matrix of order
.
When we take transpose of a matrix, then
becomes
, but for a diagonal matrix
, therefore,

We conclude that the diagonal matrix is symmetric and
is true.
Power of Diagonal Matrix
If a diagonal matrix is multiplied by itself k-times, then we can say that the matrix
is raised to the power of
.
Example #7
Let
be a diagonal matrix of order
. Let matrix
is raised to power
then
Therefore,

Invertible Diagonal Matrix
Any matrix is invertible if its determinant is not equal to 0 and it is a square matrix. The diagonal matrix is a square matrix, but it must have a non-zero entry in the main diagonal to be invertible.
If the main diagonal has a zero entry then it is a singular matrix for two reasons
- It is not a square matrix
- It has a zero determinant
Example #8
Let matrix
be a diagonal matrix of order
.
Rule 1: Diagonal matrix must be a square matrix
The last row of matrix
is zero and must be deleted. The remaining two row is not a square matrix; therefore, not invertible.
Rule 2: The determinant of matrix
should be non-zero.
The minor matrix of
is

The determinant of a diagonal matrix is
if there are non-zero elements in the main diagonal.

Inverse of a Diagonal Matrix
The inverse of a diagonal matrix can be found by using the following equation.

Example #9
Let
be a diagonal matrix of order
.
From the discussion above we know that the cofactor matrix of A is
Since, matrix
is symmetric, that is,
.
Therefore,

From the example it is clear that the inverse of a diagonal matrix contains reciprocal of each element of the diagonal matrix
.
‘Example #10
Find the inverse of following diagonal matrix .

Solution :
Given the diagonal matrix
.
We simply need to find the inverse of each diagonal element in the matrix
.
Therefore,

Important Points To Remember
Here are some important points to remember.
- diagonal addition and multiplication with another diagonal matrix is commutative.
- diagonal matrix is symmetric.
- power of diagonal matrix is power of individual diagonal entries.
- invertible diagonal matrix has non-zero diagonal entries.
- inverse of a diagonal matrix is a matrix that has inverse of each corresponding element from the diagonal matrix.