Examples on Scalar Matrix

Example 1: Calculate the determinant of a scalar matrix given below.

[Tex]A = \left[\begin{array}{ccc} -3 & 0 & 0\\ 0 & -3 & 0\\ 0 & 0 & -3 \end{array}\right] [/Tex]

Solution:

Given matrix  [Tex]A = \left[\begin{array}{ccc} -3 & 0 & 0\\ 0 & -3 & 0\\ 0 & 0 & -3 \end{array}\right] [/Tex]

|A| = −3[(−3 × −3) − 0] − 0 + 0

|A| = −3(9) = −27

Hence, the determinant of the given scalar matrix is −27.

Example 2: Give an example of a scalar matrix that has three rows and three columns.

Solution:

The order of a scalar matrix that has three rows and three columns is “3 × 3.” The matrix given below represents a scalar matrix of order “3 × 3,” where all the principal diagonal elements are equal, and the rest of the elements are zeros.

[Tex]B = \left[\begin{array}{ccc} 6 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & 6 \end{array}\right] [/Tex]

Example 3: Determine the inverse of the scalar matrix given below.

[Tex]P = \left[\begin{array}{cc} \frac{1}{2} & 0\\ 0 & \frac{1}{2} \end{array}\right] [/Tex]

Solution:

The given matrix P = [Tex]\left[\begin{array}{cc} \frac{1}{2} & 0\\ 0 & \frac{1}{2} \end{array}\right] [/Tex]

Now, P^-1 = Adj P/|P|

|P| = 1/2(1/2 − 0) − 0 = 1/4

P^-1 = [Tex]\left[\begin{array}{cc} \frac{1}{2} & 0\\ 0 & \frac{1}{2} \end{array}\right]       [/Tex]/ (1/1/4) 

P^-1 = 4 × [Tex]\left[\begin{array}{cc} \frac{1}{2} & 0\\ 0 & \frac{1}{2} \end{array}\right] [/Tex]

P^-1 = [Tex]\left[\begin{array}{cc} 2 & 0\\ 0 & 2 \end{array}\right] [/Tex]

Example 4: Find the value of (a + b + c) if the matrix given below, is a scalar matrix.

[Tex]C = \left[\begin{array}{ccc} a & 0 & 0\\ 0 & -2 & b+3\\ c-5 & 0 & -2 \end{array}\right] [/Tex]

Solution:

If the given matrix is a scalar matrix, then all its principal diagonal elements are equal, and the rest of the elements are zeros.

So, a = −2

b + 1 = 0 q = −3

c − 2 = 0 c = 5

Now, a + b + c = −2 + (−3) + 5

= −5 + 5 = 0

Hence, the value of (a + b + c) is 0 if matrix A is a scalar matrix.

Scalar matrix is a type of diagonal matrix that has all the elements the same or equal. The elements that are present other than in the diagonal are zero.

In this article, we have covered the definition of scalar matrix, its properties, formula, examples and others in detail.