If A is a square matrix of order 3,then consi | vedclass

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

Question

(A) For a square matrix $A$ of order $n=3$,the property of the adjoint matrix is $|\operatorname{Adj} A| = |A|^{n-1}$.Substituting $n=3$,we get $|\ope

Vedclass · Accepted Answer

Both $I$ and $II$

Vedclass · Answer

(A) For a square matrix $A$ of order $n=3$,the property of the adjoint matrix is $|\operatorname{Adj} A| = |A|^{n-1}$.Substituting $n=3$,we get $|\operatorname{Adj} A| = |A|^{3-1} = |A|^2$.If $|A|=0$,then $|\operatorname{Adj} A| = 0^2 = 0$. Thus,statement $I$ is true.For statement $II$,we know that $A \cdot A^{-1} = I$,where $I$ is the identity matrix.Taking the determinant on both sides,$|A \cdot A^{-1}| = |I|$.Using the property $|AB| = |A||B|$,we get $|A| \cdot |A^{-1}| = 1$.Since $|A| 
eq 0$,we can divide by $|A|$,resulting in $|A^{-1}| = \frac{1}{|A|} = |A|^{-1}$. Thus,statement $II$ is true.Therefore,both statements $I$ and $II$ are correct.

If $A$ is a square matrix of order $3$,then consider the following statements.
$I$. If $|A|=0$,then $|\operatorname{Adj} A|=0$
$II$. If $|A| \neq 0$,then $|A^{-1}|=|A|^{-1}$
Which of the above statements is/are true?

Similar Questions

If $A=\begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}$ and $B=\begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix}$,then $(A+B)^{-1} = $

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$,$10B = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{bmatrix}$ and $B$ is the inverse of matrix $A$,then $\alpha$ is equal to . . . . . . .

If $A$ is a square matrix such that $A(\operatorname{adj} A) = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$, then $\operatorname{det}(\operatorname{adj} A)$ is equal to

The inverse of a symmetric matrix is

If $A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$ such that $A^2 - 4A + 3I = 0$,where $I$ is a unit matrix of order $2$,then $A^{-1}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module

If $A$ is a square matrix of order $3$,then consider the following statements.$I$. If $|A|=0$,then $|\operatorname{Adj} A|=0$$II$. If $|A| \neq 0$,then $|A^{-1}|=|A|^{-1}$Which of the above statements is/are true?