Question

Show that the matrix $k{\text{A}}$ is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Answer 1

Hint- Here, we will be using the conditions for a matrix to be a symmetric or skew symmetric matrix.
Let A be a matrix and A’ be the transpose of the matrix A
So, matrix A is said to be symmetric matrix if ${\text{A = A'}}$ and matrix A is said to be skew symmetric matrix if ${\text{A}} = - {\text{A'}}$.
Let there be any matrix B such that ${\text{B}} = k{\text{A}}$ where $k$ is any constant.
Now for matrix B to be symmetric, \[{\text{B}} = {\text{B'}} \Rightarrow k{\text{A}} = \left( {k{\text{A}}} \right)'\]
Since for any constant $k$, \[\left( {k{\text{A}}} \right)' = k{\text{A'}} \Rightarrow k{\text{A}} = k{\text{A'}} \Rightarrow {\text{A}} = {\text{A'}}\] which is the condition for matrix A to be symmetric matrix i.e., for matrix ${\text{B}} = k{\text{A}}$ to be symmetric, matrix A should be a symmetric matrix
Now for matrix B to be skew symmetric, \[{\text{B}} = - {\text{B'}} \Rightarrow k{\text{A}} = - \left( {k{\text{A}}} \right)'\]
Since for any constant $k$, \[\left( {k{\text{A}}} \right)' = k{\text{A'}} \Rightarrow k{\text{A}} = - k{\text{A'}} \Rightarrow {\text{A}} = - {\text{A'}}\] which is the condition for matrix A to be skew symmetric matrix i.e., for matrix ${\text{B}} = k{\text{A}}$ to be skew symmetric, matrix A should be a skew symmetric matrix.
Therefore, matrix ${\text{B}} = k{\text{A}}$ is symmetric or skew symmetric according as matrix A is symmetric or skew symmetric.

Note- These types of problems can be solved by using the conditions for symmetric and skew symmetric matrices. The condition to be proved is to be simplified as much as possible