Q42 / 90 · JEE Main 2024 (08 Apr Shift 1) · Mathematics

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

  • A
    -9
  • Correct answer
    B
    -13
  • C
    -10
  • D
    -12

Solution

$\begin{aligned} & A^3-4 A^2+A+21 I=0 \\ & \operatorname{tr}(A)=4=5+6 \Rightarrow b=-1 \\ & |A|=-21 \\ & -16+a=-21 \Rightarrow a=-5 \\ & 2 a+3 b=-13\end{aligned}$

Hence, the correct answer is option: 'B'.

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