Question

If $A$ is non-singular matrix, then $A\,(adj.A)=$.
A. $A$
B. $I$
C. $|A|I$
D. $|A{{|}^{2}}I$

Answer 1

Hint: If the determinant of a square matrix is not zero that is $|A|\ne 0$ which means inverse of this matrix exists, then that matrix is termed as non-singular matrix.
The identity matrix of order $n$ can be defined as a square matrix in which elements of all the principal diagonals are one and rest of the other elements is zero.

Formula Used: \[{{A}^{-1}}=\dfrac{1}{|A|}\,adj(A)\]
\[A.{{A}^{-1}}=I\], where $I$ is identity matrix.

Complete step by step solution: We are given that $A$ is non-singular matrix, and we have to find the value of $A\,(adj.A)$. We know that according to the property of adjoint of a square matrix $A\,(adj.A)=\,(adj.A).A=|A|I$.
To derive this property we will use the formula of \[{{A}^{-1}}=\dfrac{1}{|A|}\,adj(A)\].
Now we will multiply the equation by $A$on both sides,
\[A.{{A}^{-1}}=A.\left( \dfrac{1}{|A|}\,adj(A) \right)\]
\[I=\dfrac{A.adj(A)}{|A|}\,\]
From above derived equation we can say that,
$|A|I=A\,(adj.A)$
$|A|I=\,(adj.A).A$,
Now as we have to find the value of $A\,(adj.A)$, we can say that it will be $A\,(adj.A)=|A|I$.
The value of $A\,(adj.A)$is $A\,(adj.A)=|A|I$ where$A$ is non-singular matrix.

Option ‘C’ is correct

Note: When any square matrix which has an inverse that is singular matrix is multiplied by its inverse then the product is an identity matrix that is \[A.{{A}^{-1}}=I\]. The inverse of the matrix exists only if the determinant of the matrix is a non-zero element. The non-singular matrix therefore is also called as invertible matrix.
The non-singular matrix will be always a square matrix having order of type $n\times n$ because determinants can only be calculated for the non-singular matrix.