(C) Let $A$ and $B$ be square matrices of order $n$. We know that $\text{adj}(kA) = k^{n-1} \text{adj}(A)$. For the expression $\text{trace}(\text{adj

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

$\text{trace}(\text{adj}(|A| |B| AB)) - \text{trace}(\text{adj}(|AB| BA)) = 0$

(C) Let $A$ and $B$ be square matrices of order $n$. We know that $\text{adj}(kA) = k^{n-1} \text{adj}(A)$. For the expression $\text{trace}(\text{adj}(|A||B|AB))$,we have: $\text{trace}((|A||B|)^{n-1} \text{adj}(AB)) = (|A||B|)^{n-1} \text{trace}(\text{adj}(B) \text{adj}(A))$. Similarly,for $\text{trace}(\text{adj}(|AB|BA))$,we have: $\text{trace}((|AB|)^{n-1} \text{adj}(BA)) = (|A||B|)^{n-1} \text{trace}(\text{adj}(A) \text{adj}(B))$. Since $\text{trace}(XY) = \text{trace}(YX)$ for any two matrices $X$ and $Y$,it follows that $\text{trace}(\text{adj}(B) \text{adj}(A)) = \text{trace}(\text{adj}(A) \text{adj}(B))$. Therefore,the difference is $0$. Thus,option $C$ is correct.

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

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

A
$\text{trace}(\text{adj}(AB)) = \text{adj}(\text{trace}(AB) \cdot I)$
B
$\text{trace}((A + B)(A - B)) \neq \text{trace}(A^2) - \text{trace}(B^2)$
C
$\text{trace}(\text{adj}(|A| |B| AB)) - \text{trace}(\text{adj}(|AB| BA)) = 0$
D
If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix,then $\text{trace}(AB' - BA') \neq 0$

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

Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4} \end{bmatrix}$,then the matrix $A^{31}$ is equal to

DifficultJEE MAIN 2020

View Solution

Let $A = \begin{bmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}$,where $a, b \in \mathbb{R}$. If for some $n \in \mathbb{N}$,$A^n = \begin{bmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{bmatrix}$,then $n + a + b$ is equal to:

DifficultJEE MAIN 2022

View Solution

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

MediumJEE MAIN 2021

View Solution

Match the statements given in Column $I$ with the intervals/union of intervals given in Column $II$.
Column $I$ Column $II$
$(A)$ The set $\{\operatorname{Re}(\frac{2 i z}{1-z^2}): |z|=1, z \neq \pm 1\}$ is $(p)$ $(-\infty,-1) \cup(1, \infty)$
$(B)$ The domain of $f(x)=\sin ^{-1}(\frac{8(3)^{x-2}}{1-3^{2(x-1)}})$ is $(q)$ $(-\infty, 0) \cup(0, \infty)$
$(C)$ If $f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$,then the set $\{f(\theta): 0 \leq \theta < \frac{\pi}{2}\}$ is $(r)$ $[2, \infty)$
$(D)$ If $f(x)=x^{3 / 2}(3 x-10), x \geq 0$,then $f(x)$ is increasing in $(s)$ $(-\infty,-1] \cup[1, \infty)$
$(t)$ $(-\infty, 0] \cup[2, \infty)$

DifficultIIT 2011

View Solution

If $x, y$ are any two non-zero real numbers, $a_{i j} = xi + yj$, $A = \{a_{i j}\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A = xP + yQ$, then

MediumTS EAMCET 2018

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

Column $I$	Column $II$
$(A)$ The set $\{\operatorname{Re}(\frac{2 i z}{1-z^2}): \|z\|=1, z \neq \pm 1\}$ is	$(p)$ $(-\infty,-1) \cup(1, \infty)$
$(B)$ The domain of $f(x)=\sin ^{-1}(\frac{8(3)^{x-2}}{1-3^{2(x-1)}})$ is	$(q)$ $(-\infty, 0) \cup(0, \infty)$
$(C)$ If $f(\theta)=\left\|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right\|$,then the set $\{f(\theta): 0 \leq \theta < \frac{\pi}{2}\}$ is	$(r)$ $[2, \infty)$
$(D)$ If $f(x)=x^{3 / 2}(3 x-10), x \geq 0$,then $f(x)$ is increasing in	$(s)$ $(-\infty,-1] \cup[1, \infty)$
	$(t)$ $(-\infty, 0] \cup[2, \infty)$

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

Similar Questions

Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4} \end{bmatrix}$,then the matrix $A^{31}$ is equal to

Let $A = \begin{bmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}$,where $a, b \in \mathbb{R}$. If for some $n \in \mathbb{N}$,$A^n = \begin{bmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{bmatrix}$,then $n + a + b$ is equal to:

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

If $x, y$ are any two non-zero real numbers, $a_{i j} = xi + yj$, $A = \{a_{i j}\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A = xP + yQ$, then

Vedclass Test Series

Exam Paper Generator

Online Exam Module