Question

Define a scalar matrix.

Answer 1

Hint- Here, we will proceed with the definition of the scalar matrix and then we will observe the relationship between the scalar matrix of any order and the identity matrix of that same order. Order of any matrix defines the number of rows and columns present in that matrix.

Complete step-by-step solution -
The scalar matrix is a square matrix in which all the off-diagonal elements are zero and all the on-diagonal elements are equal. We can say that a scalar matrix is a multiple of an identity matrix with any scalar quantity .
For example, $\left( {\begin{array}{*{20}{c}}
  { - 3}&0 \\
  0&{ - 3}
\end{array}} \right) = - 3{{\text{I}}_{2 \times 2}},\left( {\begin{array}{*{20}{c}}
  5&0&0 \\
  0&5&0 \\
  0&0&5
\end{array}} \right) = 5\left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right) = 5{{\text{I}}_{3 \times 3}}$ are scalar matrices.

Note- Any general ${\text{3}} \times {\text{3}}$ order scalar matrix is represented as $\left( {\begin{array}{*{20}{c}}
  x&0&0 \\
  0&x&0 \\
  0&0&x
\end{array}} \right)$ where $x$ is any number and any general $2 \times 2$ order scalar matrix is represented as $\left( {\begin{array}{*{20}{c}}
  x&0 \\
  0&x
\end{array}} \right)$ where $x$ is any number. Also, the calculation related to the evaluation of the determinant of any scalar matrix is simple because in all rows or columns of a scalar matrix there is only one non-zero element present.