Let A and B be any two n × n matrices such that following conditions hold: A B=B A and there exist

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

normal

Let $A$ and $B$ be any two $n \times n$ matrices such that the following conditions hold: $A B=B A$ and there exist positive integers $k$ and $l$ such that $A^k=I$ ( the identity matrix) and $B^l=0$ (the zero matrix). Then,

A

$A+B=I$

B

$\operatorname{det}(A B)=0$

C

$\operatorname{det}(A+B) \neq 0$

D

$(A+B)^m=0$ for some integer $m$

(KVPY-2011)

Solution

(b)

We have, $A B=B A$

$A^k=I, B^l =0$

$\left|A^k\right|=1,\left|B^l\right| =0$

$|B| =0$

$\operatorname{det}(A B)=|A||B|=0$

Standard 12

Mathematics

Share

Similar Questions

If $A$ and $B$ are square matrices of size $n \times n$ such that ${A^2} – {B^2} = \left( {A – B} \right)\left( {A + B} \right)$ ,then which ofthe following will be always true ?

easy

(AIEEE-2006)

If the matrix $A=\left(\begin{array}{cc}0 & 2 \\ K & -1\end{array}\right)$ satisfies $A\left(A^{3}+3 I\right)=2 I$ then the value of $\mathrm{K}$ is :

hard

(JEE MAIN-2021)

$A$ and $B$ be $3 \times 3$ matrices such that $AB + A + B = 0$ , then

normal

Let the determinant of a $3 \times 3$ matrix A be 6 then $B$ is a matrix defined by $B = 5{A^2}$. Then det of $B$ is

hard

Solve the equation for $x,\, y, \,z$ and $t$ if

$2\left[\begin{array}{ll}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

easy