Let A = [ begin array 35 l 1 & 2 \\0 & 1 \\end array ] and B = I + adj ( A )+( adj A ) 2 +ldots+ ( adj A ) 10 . Then, the sum of all the elements of the matrix B is :

Mathematics · Matrices

JEE Main 2024 — 4 April, Shift 2 — Question 6

Let $\text{A}=\left[ \begin{array}{*{35}{l}}1 & 2 \\0 & 1 \\\end{array} \right]$ and $\mathrm{B}=\mathrm{I}+\operatorname{adj}(\mathrm{A})+(\operatorname{adj} \mathrm{A})^{2}+\ldots+$ $(\operatorname{adj} \mathrm{A})^{10}$ .

Then, the sum of all the elements of the matrix $B$ is :

Option A:
-110
Option B:
22
Option C:
-88
Correct
Option D:
-124

Answer: C

Step-by-step solution

$Adj\left(A\right)=\begin{bmatrix}1&-2\\0&1\end{bmatrix}$ ${{(\text{AdjA})}^{2}}=\left[ \begin{matrix}1 & -4 \\0 & 1 \\\end{matrix} \right]$

${{(\text{AdjA})}^{10}}=\left[ \begin{matrix}1 & -20 \\0 & 1 \\\end{matrix} \right]$

$B=\left[ \begin{array}{*{35}{l}}1 & 0 \\0 & 1 \\\end{array} \right]+\left[ \begin{matrix}1 & -2 \\0 & 1 \\\end{matrix} \right]+\left[ \begin{matrix}1 & -4 \\0 & 1 \\\end{matrix} \right]+\ldots +\left[ \begin{matrix}1 & -20 \\0 & 1 \\\end{matrix} \right]$

$B=\left[ \begin{matrix}11 & -110 \\0 & 11 \\\end{matrix} \right]$

$\Rightarrow$ The sum of elements of $B$ $=-88$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 2
Subject: Mathematics
Chapter: Matrices
Topic: Adjoint of a Square Matrix

Let A be a 2 × 2 symmetric matrix such that A [ beginarray 35l1 \\1 \\endarray ]= [ beginarray 35l3 \\7 \\endarray ] and the determinant of…

Answer: C

Step-by-step solution

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