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| = β3[(β3 Γ β3) β 0] β 0 + 0
|A| = β3(9) = β27
Hence, the determinant of the given scalar matrix is β27.
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 β3 Γ 3.β The matrix given below represents a scalar matrix of order β3 Γ 3,β 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 β 0) β 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 Γ [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 = β2
b + 1 = 0 q = β3
c β 2 = 0 c = 5
Now, a + b + c = β2 + (β3) + 5
= β5 + 5 = 0
Hence, the value of (a + b + c) is 0 if matrix A is a scalar matrix.
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.
Table of Content
- Definition of Scalar Matrix
- Condition for a Scalar Matrix
- Examples of Scalar Matrix
- Properties of a Scalar Matrix
- Operation on Scaler Matrix
- Examples on Scalar Matrix
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