Question

Let $A$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either $0$ or $1$ . Five of these entries are $1$ and four of them are $0$.The number of matrices in $A$ is
The number of matrices $A$ in for which the system of linear equations $A\begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}1 \\0 \\0\end{bmatrix}$ is inconsistent, is

• A. 0
• B. more than 2
• C. 2
• D. 1

Answer 1

Answer: (B)

The six matrix $A$ for which $| A |=0$ are
$\begin{bmatrix}0&0&1\\ 0&0&1\\ 1&1&1\end{bmatrix} \Rightarrow $ inconsistent.
$\begin{bmatrix}0&1&0\\ 1&1&1\\ 0&1&0\end{bmatrix} \Rightarrow $ inconsistent
$ \begin{bmatrix}1&1&1\\ 1&0&0\\ 1&0&0\end{bmatrix} \Rightarrow $ infinite solutions
$ \begin{bmatrix}1&1&1\\ 1&1&0\\ 0&0&1\end{bmatrix} \Rightarrow $ inconsistent
$\begin{bmatrix}1&0&1\\ 0&1&0\\ 1&0&1\end{bmatrix} \Rightarrow $ inconsistent
$\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&1&1\end{bmatrix} \Rightarrow $ infinite solutions