Home  >>  CBSE XII  >>  Math  >>  Matrices

# If the matrix $\begin{bmatrix}0 & a & 3\\2 & b & -1\\c & 1 & 0\end{bmatrix}\;$is a skew symmetric matrix,find the values of a,b and c.

Toolbox:
• A square matrix A=[a$_{ij}$] is said to be skew symmetric if A'=-A that is $[a_{ij}]= -[a_{ji}]$ for all possible value of i and j.
• The scalar multiplication $cA$ of a matrix $A$ and a number $c$ (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of $A$ by $c$.
The given matrix is a skew symmetric matrix,
Let $A=\begin{bmatrix}0 & a & 3\\2 & b & -1\\c & 1 & 0\end{bmatrix}$
$A^T=\begin{bmatrix}0 & 2 & c\\a & b & 1\\3 & -1 & 0\end{bmatrix}$
Skew symmetric matrix $\Rightarrow A'=-A.$
$\begin{bmatrix}0 & 2 & c\\a & b & 1\\3 &- 1 & 0\end{bmatrix}=(-1)\begin{bmatrix}0 & a & 3\\2 & b & -1\\c & 1 & 0\end{bmatrix}$
$\begin{bmatrix}0 & 2 & c\\a & b & 1\\3 &- 1 & 0\end{bmatrix}=\begin{bmatrix}0 & -a & -3\\-2 & -b & 1\\-c & -1 & 0\end{bmatrix}$
The given two matrices are equal hence their corresponding element should be equal.
$\Rightarrow$ From second column we have,
-a=2.
a=-2.
$\Rightarrow$ From third column we have,
c=-3.
b=-b
From the property we have all the diagonal element of skew matrix are zero.
$\Rightarrow$ b=0.