Let A = <math><mrow><mrow><mo>|</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mi>a</mi></mtd><mtd columnalign="center"><mi>a</mi></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mi>b</mi></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd></mtr></mtable></mrow><mo>|</mo></mrow></mrow></math> , a, b <math><mrow><mo>∈</mo></mrow></math> R. If for some n <math><mrow><mo>∈</mo></mrow></math> N, An = <math><mrow><mrow><mo>|</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mn>4</mn><mn>8</mn></mtd><mtd columnalign="center"><mn>2</mn><mn>1</mn><mn>6</mn><mn>0</mn></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mn>9</mn><mn>6</mn></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd></mtr></mtable></mrow><mo>|</mo></mrow></mrow></math> then n + a + b is equal to

A=[100010001]+[0aa00b000]=l+BB2=[0aa00b000]×[0aa00b000]=[00ab000000]B3 = 0∴An=(1+B)n=nC0l+nC1B+nC2B2+nC3B3+....On comparing we get na = 48, nb = 96 and na + n(n−1)2ab=2160⇒a=4,n=12 and b=8⇒n+a+b=24

Ask & Answer Question

Maths Class 12th Maths Matrices

Let A = $| \begin{matrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} |$ , a, b $\in$ R. If for some n $\in$ N, Aⁿ = $| \begin{matrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{matrix} |$ then n + a + b is equal to

6 Views | Posted 3 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

3 months ago

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] + [\begin{matrix} 0 & a & a \\ 0 & 0 & b \\ 0 & 0 & 0 \end{matrix}] = l + B$

$B^{2} = [\begin{matrix} 0 & a & a \\ 0 & 0 & b \\ 0 & 0 & 0 \end{matrix}] \times [\begin{matrix} 0 & a & a \\ 0 & 0 & b \\ 0 & 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 0 & a b \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

B³ = 0

$∴ A^{n} = {(1 + B)}^{n} =^{n} C_{0} l +^{n} C_{1} B +^{n} C_{2} B^{2} +^{n} C_{3} B^{3} + . . . .$

On comparing we get na = 48, nb = 96 and na + $\frac{n (n - 1)}{2} a b = 2160$

$\Rightarrow a = 4, n = 12 a n d b = 8$

$\Rightarrow n + a + b = 24$

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
679k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

Let A = $| \begin{matrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} |$ , a, b $\in$ R. If for some n $\in$ N, Aⁿ = $| \begin{matrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{matrix} |$ then n + a + b is equal to

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let A = |1aa01b001| , a, b ∈ R. If for some n ∈ N, An = |14821600196001| then n + a + b is equal to

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let A = $| \begin{matrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} |$ , a, b $\in$ R. If for some n $\in$ N, Aⁿ = $| \begin{matrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{matrix} |$ then n + a + b is equal to