Question

If A=$\left|\begin{array}{cc}2& 3\\ 5& 1\end{array}\right|,B=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$ and AB= -13I then find the value of a+b-c+d

• A. 2
• B. 3
• C. 4
• D. 1

Answer 1

Answer: (D)

1

Given, $AB=-13I$

$\Rightarrow \left|\begin{array}{cc}2& 3\\ 5& 1\end{array}\right|\times \left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=-13\times \left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|$

$\Rightarrow \left|\begin{array}{cc}2\left(a\right)+3\left(c\right)& 2\left(b\right)+3\left(d\right)\\ 5\left(a\right)+1\left(c\right)& 5\left(b\right)+1\left(d\right)\end{array}\right|=\left|\begin{array}{cc}-13& 0\\ 0& -13\end{array}\right|$

By comparing

$\Rightarrow 2a+3c=-13$ --- (1)

$\Rightarrow 2b+3d=0$ --- (2)

$\Rightarrow 5a+c=-13$ --- (3)

$\Rightarrow 5b+d=0$ --- (4)

Solving equations 1 and 3, we get $a=-2,c=-3$

Solving equations 2 and 4, we get $b=0,d=0$

So, $a+b-c+d=-2+0+3+0=1$