JEE Advanced 2019 · Paper 1, Mathematics question 6

Find a and b. Entry (1,1) of adj M is the cofactor $2-3b=-1$, so $b=1$. Entry (1,2) is $-(1-ab)=1$, so $ab=2$ and $a=2$.

Determinant. $\det M=-1(1-9)+2(b-6)=8-10=-2$.

The options.

• (A) $a+b=3$. True.
• (B) $(\operatorname{adj}M)^{-1}=\operatorname{adj}(M^{-1})=\dfrac M{\det M}$, so the sum is $\dfrac{2M}{-2}=-M$. True.
• (C) $\det(\operatorname{adj}M^2)=(\det M^2)^2=(\det M)^4=16\ne81$. False.
• (D)
  $$\begin{bmatrix}\alpha\\\beta\\\gamma\end{bmatrix}=\frac{\operatorname{adj}M}{\det M}\begin{bmatrix}1\\2\\3\end{bmatrix}=-\frac12\begin{bmatrix}-2\\2\\-2\end{bmatrix}=\begin{bmatrix}1\\-1\\1\end{bmatrix},$$
  so $\alpha-\beta+\gamma=3$. True.

(A), (B), (D)