(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

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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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If $\text{adj} \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n \end{bmatrix}$,then $m+n=$

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$A$ and $B$ are $n$-rowed square matrices such that $AB = O$ and $B$ is non-singular. Then:

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If $|\operatorname{Adj} A|=x$ and $|\operatorname{Adj} B|=y$,then $\left|(\operatorname{Adj}(AB))^{-1}\right|=$

EasyTS EAMCET 2024

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If the determinant of the adjoint of a (real) matrix of order $3$ is $25$,then the determinant of the inverse of the matrix is

EasyKCET 2013

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If $x, y, z$ are nonzero real numbers,then the inverse of matrix $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is

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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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If $\text{adj} \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n \end{bmatrix}$,then $m+n=$

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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