(C) Given that $A=B+C$,$BC=CB$,and $C^2=0$. Since $B$ and $C$ commute,we can use the Binomial Theorem for matrices: $A^k = (B+C)^k = \sum_{r=0}^{k} \b

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(C) Given that $A=B+C$,$BC=CB$,and $C^2=0$. Since $B$ and $C$ commute,we can use the Binomial Theorem for matrices: $A^k = (B+C)^k = \sum_{r=0}^{k} \binom{k}{r} B^{k-r} C^r$. Since $C^2=0$,all higher powers $C^r=0$ for $r \ge 2$. Thus,$A^k = \binom{k}{0} B^k C^0 + \binom{k}{1} B^{k-1} C^1 = B^k + k B^{k-1} C = B^{k-1}(B+kC)$. Setting $k=2021$,we get: $A^{2021} = B^{2021-1}(B+2021C) = B^{2020}(B+2021C)$. Therefore,$B^{2020}[B+(2021)C] = A^{2021}$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

Medium

Let $B$ and $C$ be $n \times n$ matrices such that $A=B+C$,$BC=CB$,and $C^2=0$ (where $0$ is the null matrix). Then,$B^{2020}[B+(2021)C]=$

TS EAMCET 2020 All TS EAMCET

A
$A^{2020}$
B
Null matrix of order $n \times n$
C
$A^{2021}$
D
$B^{2021}$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

3 and 4 .Determinants and Matrices

Subtopic

Mix Examples-Determinants and Matrices

Previous Year

TS EAMCET 2020 Paper All TS EAMCET years →

If $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $\alpha, \beta, \gamma$ are the roots of the characteristic equation $|A - xI| = 0$,then $\alpha^2 + \beta^2 + \gamma^2 = $

MediumAP EAMCET 2025

View Solution

The number of $3 \times 3$ non-singular matrices,with four entries as $1$ and all other entries as $0$,is

MediumAIEEE 2010

View Solution

Let $A = \begin{bmatrix} 1 & 2 \\ -2 & -5 \end{bmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^{2} + \beta A = 2I$. Then $\alpha + \beta$ is equal to -

MediumJEE MAIN 2022

View Solution

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & 0 & k \end{bmatrix}$ and $f(x) = x^3 - 2x^2 - \alpha x + \beta = 0$. If $A$ satisfies $f(A) = 0$,then:

View Solution

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$. Then $3a^2 + 4b^2$ is equal to:

DifficultJEE MAIN 2023

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

Let $B$ and $C$ be $n \times n$ matrices such that $A=B+C$,$BC=CB$,and $C^2=0$ (where $0$ is the null matrix). Then,$B^{2020}[B+(2021)C]=$

Similar Questions

If $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $\alpha, \beta, \gamma$ are the roots of the characteristic equation $|A - xI| = 0$,then $\alpha^2 + \beta^2 + \gamma^2 = $

The number of $3 \times 3$ non-singular matrices,with four entries as $1$ and all other entries as $0$,is

Let $A = \begin{bmatrix} 1 & 2 \\ -2 & -5 \end{bmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^{2} + \beta A = 2I$. Then $\alpha + \beta$ is equal to -

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & 0 & k \end{bmatrix}$ and $f(x) = x^3 - 2x^2 - \alpha x + \beta = 0$. If $A$ satisfies $f(A) = 0$,then:

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$. Then $3a^2 + 4b^2$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module