What is Scalar Matrix? Examples & Properties

A scalar matrix is a special case of a diagonal matrix, in which all diagonal elements are equal and all off-diagonal elements are zero.

A scalar matrix is always a square matrix and hence the size of this matrix will be n x n.

Condition for Scalar Matrix

Let M is a square matrix having ‘i’ number of rows and ‘j’ number of columns. So matrix M to be a scalar matrix, the following two conditions must be satisfied. ‘Mij‘ represents the element at row number ‘i’ and column number ‘j’.

1. Mij = k    for  i = j and k ≠ 0

where i = j = 0, 1, 2, …….., n

2. Mij = 0     for  i ≠ j

where i = j = 0, 1, 2, …….., n

Scalar Matrix Examples

In this section, we will see some examples of a scalar matrix.

Example-1: It is a 2 x 2 matrix.

    \[\begin{bmatrix} 3 &0 \\ 0& 3 \end{bmatrix}\]

Let check the condition to be the scalar matrix. We can easily see that the two diagonal elements are equal and equal to ‘3’. The off-diagonal elements are zero. Since this matrix follows the above discussed two conditions. Hence it is a scalar condition.

Example-2: It is a 3 x 3 matrix.

    \[\begin{bmatrix} -5 &0 &0 \\ 0& -5 &0 \\ 0&0 & -5 \end{bmatrix}\]

It is also a scalar matrix because its all diagonal elements are equal (have value equal to ‘-5’) and off-diagonal elements are zero.

Is identity matrix a scalar matrix?

Many of us confuse whether an identity matrix is a scalar matrix or not. Hope, you will get an answer to this question in this article. Let us take an example of a 3 x 3 identity matrix.

    \[\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0& 0 & 1 \end{bmatrix}\]

Now applying the conditions of the scalar matrix that we have studied in the starting of this article. We can easily see that it is a square matrix. Secondly, its all diagonal elements are equal (i.e. equal to ‘1’) and its off-diagonal elements are zero. Hence we can say that an identity matrix is a scalar matrix.

Can a zero matrix be called a scalar matrix?

No. We can’t say zero matrix is not a scalar matrix because its all diagonal elements are zero.

What is the determinant of a scalar matrix?

Suppose we have a scalar matrix P of size n x n. Let the value of all its diagonal elements are A. Then the determinant of a scalar matrix P will be equal to An.

For example,

    \[P=\begin{bmatrix} 5 &0 &0 \\ 0& 5 &0 \\ 0&0 & 5 \end{bmatrix}\]

Here in this example, matrix P has a size of 3 x 3. Therefore, n = 3 and A = 5.

The determinant of matrix P = 5= 125

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