jee-advanced 2021 Q19

jee-advanced · India · paper1 4 marks 3x3 Matrices Determinant of Parametric or Structured Matrix
Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?
(A) $(a+b)^2 = 9$
(B) $\det(\text{adj}(M^2)) = 81$
(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
(D) $\det(\text{adj}(2M)) = 2^8$
Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$

where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?

(A) $(a+b)^2 = 9$

(B) $\det(\text{adj}(M^2)) = 81$

(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$

(D) $\det(\text{adj}(2M)) = 2^8$