Question

Assertion :Suppose $X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ satisfies the equation $X^2-4X+3I=0$.

If $a+d\ne 4$, then there are just two such matrix $X$.

Reason: If $a+d=4$, then there are infinite number of matrices $X$, satisfying $X^2-4X+3I=0$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$X^2-4X+3I=0$

$X^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& cb+d^2\end{array}\right]$

$X^2-4X+3I=0$

$\left[\begin{array}{cc}{a}^2+bc& b(a+d)\\ c(a+d)& bc+d^2\end{array}\right]-\left[\begin{array}{cc}4a& 4b\\ 4c& 4d\end{array}\right]+\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$

If $a+d\ne 4$

Then $det\ne 0$

It has $2$ solutions

But if $a+d=4△=0$

Infinite solutions

Option B