Let A be a square matrix all of whose entries | vedclass

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

Question

(C) The inverse of a square matrix $A$ is given by the formula: $A^{-1} = \frac{1}{\det(A)} 	ext{adj}(A)$. Since $A$ is a square matrix with integer

Vedclass · Accepted Answer

If $\det(A) = \pm 1$ then $A^{-1}$ exists and its entries are integers.

Vedclass · Answer

(C) The inverse of a square matrix $A$ is given by the formula: $A^{-1} = \frac{1}{\det(A)} 	ext{adj}(A)$. Since $A$ is a square matrix with integer entries,the adjoint matrix $	ext{adj}(A)$ also consists entirely of integers (as it is the transpose of the cofactor matrix). If $\det(A) = \pm 1$,then $A^{-1} = \frac{1}{\pm 1} 	ext{adj}(A) = \pm 	ext{adj}(A)$. Since $	ext{adj}(A)$ contains only integers,$\pm 	ext{adj}(A)$ also contains only integers. Therefore,if $\det(A) = \pm 1$,$A^{-1}$ exists and all its entries are integers.

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true $?$

Similar Questions

${\left[ {\begin{array}{*{20}{c}}{ - 6}&5\\{ - 7}&6\end{array}} \right]^{ - 1}}$ =

If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right]$,$10 B=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right]$ and $B$ is the inverse of $A$,then the value of $\alpha$ is

If possible,using elementary row transformations,find the inverse of the following matrix:
$\left[\begin{array}{ccc}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$

If $F(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$,where $\alpha \in \mathbb{R}$,then $[F(\alpha)]^{-1}$ is equal to:

If $A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}_{3 \times 3}$,then $A^{-1} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module