Let A= [ begin array 35 l 2 & a & 0 \\1 & 3 & 1 \\0 & 5 & b \\end array ] .If A 3 =4 A 2 -A-21 I , where I is the identity matrix of order 3 × 3 , then 2 a+3 b is equal to :

Mathematics · Matrices

JEE Main 2024 — 8 April, Shift 1 — Question 8

Let $A=\left[ \begin{array}{*{35}{l}}2 & a & 0 \\1 & 3 & 1 \\0 & 5 & b \\\end{array} \right]$ .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 :

Option A:
$-10$
Option B:
$-13$
Correct
Option C:
$-9$
Option D:
$-12$

Answer: B

Step-by-step solution

$A^{3}-4 A^{2}+A+21 I=0$

$\operatorname{tr}(A)=4$ (from equation)

$2+3+b =4\Rightarrow b=-1$

$|\mathrm{A}|=-21$ (from equation)

$-16+a=-21 \Rightarrow a=-5$

$\boxed{2 a+3 b=-13}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 8 April, Shift 1
Subject: Mathematics
Chapter: Matrices
Topic: Characteristic Equation & roots,application of cayley - hamilton theorem

Let A= [ beginmatrix2 & -1 \\1 & 1 \\endmatrix ] . If the sum of the diagonal elements of A^13 is 3^n , then n is equal to \ \ \ \

Answer: B

Step-by-step solution

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