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 = 53 = 125
