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 $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ 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
$\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]\Rightarrow$ inconsistent.
$\left[\begin{array}{ccc}0& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right]\Rightarrow$ inconsistent
$\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]\Rightarrow$ infinite solutions
$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]\Rightarrow$ inconsistent
$\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]\Rightarrow$ inconsistent
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 1& 1\end{array}\right]\Rightarrow$ infinite solutions