Discussion Forum



1)

Let M be a 2x2 symmetric matrix with integer entries. Then, M is invertible, if


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 diagonal

D) the product of entries in the main diagonal of M is not the square of an integer

Answer:

Option C,D

Explanation:

Plan  A square matrix M  is invertible if det(M)   or |M|  $\neq$ 0,

  Let   $M=\begin{bmatrix}a & b \\b & c \end{bmatrix}$

 (a) Given that    $\begin{bmatrix}a  \\b  \end{bmatrix}=\begin{bmatrix}b  \\c  \end{bmatrix}$

 $\Rightarrow$         a=b=c= $\alpha$              (let)

$\Rightarrow$             M= $\begin{bmatrix}\alpha & \alpha \\\alpha & \alpha \end{bmatrix}$

$\Rightarrow$                       |M|=0

$\Rightarrow$                   Mis non-invertible

(b) Given that [b c]=[a b]

$\Rightarrow$                     a=b=c =  $\alpha$              (let)

                   Again  |M|=0

$\Rightarrow$       M is non-invertible

 (c)    As given   $M=\begin{bmatrix}a & 0 \\0 & c \end{bmatrix}$

  $\Rightarrow$                $|M|=ac\neq0$

                                      (   $\because$   a and c are non-zero)

$\Rightarrow$                 Mis invertiable

(d)        $M=\begin{bmatrix}a & b \\b & c \end{bmatrix}\Rightarrow|M|=ac-b^{2}\neq 0$

  $\because$ ac is not equal to square of an integer.

 $\because$   M is invertiable