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:

  1. 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.



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