Which of following statement is correct about two square matrices A and B of same order

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

normal

Which of the following statement is correct about two square matrices $A$ and $B$ of same order

A

trace $(Adj(AB))$ = $Adj(trace (AB).I)$

B

trace $((A + B) (A -B)) \neq trace (A^2) -trace(B^2)$

C

trace $(Adj(|A| |B| AB))$ -trace $(Adj(|AB| BA))$ = $0$

D

If $A$ is symmetric matrix and $B$ is skew symmetric matrix then trace $(AB' -BA')\neq 0$

Solution

$\operatorname{trace}(\operatorname{adj\,}(|\mathrm{A} \| \mathrm{B}| \mathrm{AB}))$

$=\operatorname{trace}\left(|\mathrm{A}|^{\mathrm{n}-1}|\mathrm{B}|^{\mathrm{n}-1}(\mathrm{Adj\,} \mathrm{B})(\mathrm{Adj\,} \mathrm{A})\right)$

$\operatorname{trace}(\operatorname{Adj\,}(|\mathrm{AB}| \mathrm{BA}))$

$=\operatorname{trace}\left(|A|^{n-1}|B|^{n-1}(A d j \,A) \cdot(A d j \,B)\right)$

Standard 12

Mathematics

Share

Similar Questions

Let $A$ be a square matrix of order $2$ such that $|A|=2$ and the sum of its diagonal elements is $-3$ . If the points $(x, y)$ satisfying $A^2+x A+y I=0$ lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is e and the length of the latus rectum is $\ell$, then $\mathrm{e}^4+\ell^4$ is equal to………………………

medium

(JEE MAIN-2024)

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A),$ the sum of diagonal entries of $a$. Assume that ${A^2} = I$ .

Statement $-1 :$ If $A \ne I,A \ne – I$ then $\det \left( A \right) = – 1$

Statement $-2 :$ If $A \ne I,A \ne – I$ then ${\rm{tr}}\left( A \right) \ne 0$

medium

(AIEEE-2008)

Matrix $\left[ {\begin{array}{*{20}{c}}0&{ – 4}&1\\4&0&{ – 5}\\{ – 1}&5&0\end{array}} \right]$is

easy

If $A=$ $\left[ {\begin{array}{*{20}{c}}{5a}&{ – b}\\3&2\end{array}} \right]$ and $A\;adj\;A = A\;{A^T},$ then $5a+b $ to :

medium

(JEE MAIN-2016)

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$, where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that

normal

(IIT-2012)