(B) For a square matrix $A$ of order $n$,the following statements are equivalent:$1$. $A$ is invertible.$2$. There exists a matrix $B$ such that $AB =

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$I$,$II$ and $III$ are equivalent statements

(B) For a square matrix $A$ of order $n$,the following statements are equivalent:$1$. $A$ is invertible.$2$. There exists a matrix $B$ such that $AB = I_n$.$3$. There exists a matrix $C$ such that $CA = I_n$.If $AB = I_3$,then multiplying by $A^{-1}$ on the right gives $A = I_3 B^{-1}$,which implies $B = A^{-1}$.Similarly,if $CA = I_3$,then $C = A^{-1}$.Thus,$B = C = A^{-1}$.Since all three statements are equivalent,the correct option is $B$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Adjoint and inverse of matrices

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Let $A$ be a square matrix of order $3$. Choose the correct option regarding the following statements:
$I$. There exists a matrix $B$ of order $3$ such that $AB = I_3$
$II$. There exists a matrix $C$ of order $3$ such that $CA = I_3$
$III$. $A$ is invertible

AP EAMCET 2020 All AP EAMCET

A
Only $III$ implies $I$ and $II$
B
$I$,$II$ and $III$ are equivalent statements
C
In $I$ and $II$,$B$ can be different from $C$
D
None of the above

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3 and 4 .Determinants and Matrices

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Adjoint and inverse of matrices

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Find the inverse of the matrix,if it exists: $\left[\begin{array}{ccc}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$

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The inverse of the matrix $\left[ {\begin{array}{*{20}{c}}3&{ - 2}&{ - 1}\\{ - 4}&1&{ - 1}\\2&0&1\end{array}} \right]$ is

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If $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$,then $B^{-1} A^{-1} = $

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Let $A$ be a square matrix of order $3$. Choose the correct option regarding the following statements:$I$. There exists a matrix $B$ of order $3$ such that $AB = I_3$$II$. There exists a matrix $C$ of order $3$ such that $CA = I_3$$III$. $A$ is invertible

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Find the inverse matrix of $\begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix}$.

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Let $A$ be a square matrix of order $3$. Choose the correct option regarding the following statements:
$I$. There exists a matrix $B$ of order $3$ such that $AB = I_3$
$II$. There exists a matrix $C$ of order $3$ such that $CA = I_3$
$III$. $A$ is invertible