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.

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### 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.