If \( A=\begin{bmatrix}3 & 5\\a & b\end{bmatrix} \) and \( \mathrm{adj}\,A=\begin{bmatrix}2b-1 & -5\\a+b & c\end{bmatrix} \), then the value of \( 4(a+b-c)(a-b+c) \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → −15
For a 2 × 2 matrix, the adjoint is found by swapping the diagonal entries and changing the signs of the other two:
\( \displaystyle \mathrm{adj}\,A=\begin{bmatrix}b & -5\\-a & 3\end{bmatrix} \)
Comparing with the given adjoint entry by entry:
\( \displaystyle b=2b-1\ \Rightarrow\ b=1 \)
\( \displaystyle -a=a+b\ \Rightarrow\ -2a=1 \)
\( \displaystyle a=-\frac{1}{2} \)
\( \displaystyle c=3 \)
Then
\( \displaystyle 4(a+b-c)(a-b+c)=4\left(-\frac{1}{2}+1-3\right)\left(-\frac{1}{2}-1+3\right) \)
\( \displaystyle =4\left(-\frac{5}{2}\right)\left(\frac{3}{2}\right)=-15 \)