Matrices and Determinants 3 Question 12
14. Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then, $M$ is invertible, if
(2014 Adv.)
(a) the first column of $M$ is the transpose of the second row of $M$
(b) the second row of $M$ is the transpose of the first column of $M$
(c) $M$ is a diagonal matrix with non-zero entries in the main digonal
(d) the product of entries in the main diagonal of $\mathrm{M}$ is not the square of an integer
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Answer:
Correct Answer: 14. (c,d)
Solution:
- PLAN: A square matrix $M$ is invertible, if dem $(M)$ or $|M| \neq 0$.
Let $\quad M=$ $ \begin{bmatrix} a & b \\ b & c\end{bmatrix}$
(a) Given, $\frac{a}{b}={ }_{c}^{b} \Rightarrow a=b=c=\alpha$
$\Rightarrow \quad M=\begin{bmatrix}\alpha & \alpha \\ \alpha & \alpha\end{bmatrix} \Rightarrow|M|=0 \Rightarrow M$ is non-invertible.
(b) Given, $[b c]=[a b]$
$\Rightarrow \quad a=b=c=\alpha$
Again, $|M|=0$
$\Rightarrow M$ is non-invertible.
(c) As given $M=\begin{bmatrix}a & 0 \\ 0 & c\end{bmatrix} \Rightarrow|M|=a c \neq 0$
$\Rightarrow M$ is invertible.
$[\because a$ and $c$ are non-zero]
(d) $M=\begin{bmatrix} a & b \\ b & c\end{bmatrix} \Rightarrow|M|=a c-b^{2} \neq 0$
$\because \quad a c$ is not equal to square of an integer.
$M$ is invertible.
