Question

If A = $\left[ \begin{matrix}
   1 & 0 & 1 \\
   0 & 1 & 1 \\
   1 & 0 & 0 \\
\end{matrix} \right]$ then A is
A . symmetric
B . Skew- symmetric
C . Non- singular
D . Singular

Answer 1

Hint: In this question, we have given a matrix A and we have to find out the option which follows the matrix A. For this, we use the properties of the given options. First we check whether it is symmetric or non- symmetric. Then we find out the determinant to check whether the matrix is singular or non- singular or mark the option accordingly.

Complete Step- by- step Solution:
Given matrix is A = $\left[ \begin{matrix}
   1 & 0 & 1 \\
   0 & 1 & 1 \\
   1 & 0 & 0 \\
\end{matrix} \right]$
Given matrix is of $3\times 3$ order.
First we check whether A is symmetric or skew symmetric.
${{A}^{T}}=\left[ \begin{matrix}
   1 & 0 & 1 \\
   0 & 1 & 0 \\
   1 & 1 & 0 \\
\end{matrix} \right]$
$A\ne {{A}^{T}}$
It is not a symmetric matrix
And ${{A}^{T}}\ne -A$
Hence it is not a skew symmetric matrix
Now we check whether it is singular or non singular matrix
For this we find out the determinant
$|A|=1[1-0]-0[0-1]+1[0-1]$
$|A|=1-1$
$|A|=0$
Hence, it is a singular matrix.

Thus, Option (D) is correct.

Note: Students make mistakes in finding out the transpose of a matrix. They get confused while finding the transpose. Transpose means interchanging of rows and columns. Remember that in transpose, the first row becomes the first column, the second row becomes the second column and the third row becomes the third column. If the transpose of matrix A, ${{A}^{T}}=-A$ then we call it a skew- symmetric matrix.