Question

Let $S$ be the set of all real matrices, $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ such that $a+d=2$ and $A^T=A^2-2A$. Then $S$ -

• A. Has exactly four elements
• B. Is an empty one element
• C. Has exactly one element
• D. Has exactly two element

Answer 1

Answer: (A)

Has exactly four elements

We have,

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

And $a+d=2$

$A^T=A^2-2A$

$A^T=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$=$A.A$=$A^2$=$\left[\begin{array}{cc}{a}^2-bc& ab-bd\\ ca-dc& bc-d^2\end{array}\right]$-$\left[\begin{array}{cc}2a& 2b\\ 2c& 2d\end{array}\right]$

$=\left[\begin{array}{cc}{a}^2-bc-2a& ab-bd-2b\\ ca-dc-2c& bc-d^2-2d\end{array}\right]$

Hence, this set has exactly four set.