Question

Let a matrix $A=\left[\begin{array}{cc}\cos x& \sin x\\ \tan x& \cot x\end{array}\right],$ then which of the following statement(s) is(are) true for atleast one value of $x\in [0,\frac{\pi }{2}]$ ?
(where $C_{ij}$ is co-factor of element $\left[a_{ij}\right]$)

• A. $C_{11}=1$
• B. $C_{12}=1$
• C. $C_{21}=1$
• D. $C_{22}=1$

Answer 1

Answer: (A), (D)

$C_{22}=1$

Given : $A=\left[\begin{array}{cc}\cos x& \sin x\\ \tan x& \cot x\end{array}\right]$
$\Rightarrow M_{11}=\cot x,M_{12}=\tan x,M_{21}=\sin x,M_{22}=\cos x$ and
we know, $C_{ij}=(-1{)}^{i+i}{M}_{ij}$
So,
$C_{11}=\cot x=1$ at $x=n\pi +\frac{\pi }{4}$
For $n=0,x\in [0,\frac{\pi }{2}]$
$C_{12}=-\tan x=1$ at $x=n\pi -\frac{\pi }{4}$
No possible value for $x\in [0,\frac{\pi }{2}]$
$C_{21}=-\sin x=1$ at $x=2n\pi -\frac{\pi }{2}$
No possible value for $x\in [0,\frac{\pi }{2}]$
$C_{22}=\cos x=1$ at $x=2n\pi$
For $n=0,x\in [0,\frac{\pi }{2}]$
(where $n=0,\pm 1,\pm 2,\cdots$)