Question

If the matrix $A=\left[ \begin{matrix} 0 & a & 3 \\\ 2 & b & -1 \\\ c & 1 & 0 \\\ \end{matrix} \right]$ is skew symmetric, then find the values of a, b and c?

Answer 1

We start solving the problem by recalling the definitions of skew symmetric matrix and transpose of the matrix. We then find the transpose of the given matrix and use the fact that for a skew symmetric matrix ${{A}^{T}}=-A$. We then make required calculations and equate the corresponding elements on both sides to get the required values of a, b and c.

Complete step-by-step answer :
According the problem, we are given matrix A as $\left[ \begin{matrix} 0 & a & 3 \\\ 2 & b & -1 \\\ c & 1 & 0 \\\ \end{matrix} \right]$. We need to find the values of a, b and c if the matrix A is given as skew symmetric.
Before solving for the values of a, b and c, we first recall the definition of skew symmetric matrix.
We know that a square matrix is defined as a skew symmetric matrix if the transpose of the matrix is equal to the negative of the matrix i.e., ${{A}^{T}}=-A$.
So, let us first transpose of the given matrix A. We know that the transpose of a matrix is formed by interchanging the rows with columns of given matrix. We use this to find the transpose of the matrix A.
So, we get ${{A}^{T}}={{\left[ \begin{matrix} 0 & a & 3 \\\ 2 & b & -1 \\\ c & 1 & 0 \\\ \end{matrix} \right]}^{T}}$.
$\Rightarrow {{A}^{T}}=\left[ \begin{matrix} 0 & 2 & c \\\ a & b & 1 \\\ 3 & -1 & 0 \\\ \end{matrix} \right]$.
But according to the problem, we have given that the matrix is skew symmetric. So, we have ${{A}^{T}}=-A$.
$\Rightarrow \left[ \begin{matrix} 0 & 2 & c \\\ a & b & 1 \\\ 3 & -1 & 0 \\\ \end{matrix} \right]=-\left[ \begin{matrix} 0 & a & 3 \\\ 2 & b & -1 \\\ c & 1 & 0 \\\ \end{matrix} \right]$.
$\Rightarrow \left[ \begin{matrix} 0 & 2 & c \\\ a & b & 1 \\\ 3 & -1 & 0 \\\ \end{matrix} \right]=\left[ \begin{matrix} -0 & -a & -3 \\\ -2 & -b & -\left( -1 \right) \\\ -c & -1 & -0 \\\ \end{matrix} \right]$.
$\Rightarrow \left[ \begin{matrix} 0 & 2 & c \\\ a & b & 1 \\\ 3 & -1 & 0 \\\ \end{matrix} \right]=\left[ \begin{matrix} 0 & -a & -3 \\\ -2 & -b & 1 \\\ -c & -1 & 0 \\\ \end{matrix} \right]$.
We know that if two matrices are equal, then the elements in the corresponding places are also equal.
So, we get $a=-2$, $c=-3$ and $b=-b$.
$\Rightarrow b+b=0$.
$\Rightarrow 2b=0$.
$\Rightarrow b=0$.
So, we have found the values of a, b and c as –2, 0 and –3.
∴ The values of a, b and c as –2, 0 and –3.

Note : After finding the values of a, b and c, we can see that every element in the principal diagonal of the matrix is zero. We should know that all skew-symmetric matrices are square matrices, but all square matrices are not skew symmetric matrices. We can also find the determinant of the matrix after finding the values of a, b and c. Similarly, we can expect problems to find the determinant of skew symmetric matrices.