Question

The value of $${\left(AdjA\right)}^{-1}$$is equal to
(a) $$Adj\left(A^{-1}\right)$$
(b) $$Adj[-A]$$
(c) $${\left(AdjA\right)}^T$$
(d) $$Adj\left(A^T\right)$$

Answer 1

Hint: In this question, from the given condition we need to use the properties of adjoint of a square matrix and properties of inverse of a square matrix to convert it accordingly. Here, we first write the given condition using the formula $$A^{-1}={\displaystyle \frac{1}{\left|A\right|}}\left(AdjA\right)$$. Then we need to simplify it using the properties of adjoint matrix given by $$Adj\left(AdjA\right)={\left|A\right|}^{n-2}A$$ and $$\left|AdjA\right|={\left|A\right|}^{n-1}$$. Now, on simplifying it further and converting it accordingly gives the result.

Complete step by step solution:
ADJOINT of a MATRIX:
Adjoint of a matrix is the transpose of the matrix of cofactors of the given matrix
Properties of adjoint of a square matrix are given by
$$Adj\left(AdjA\right)={\left|A\right|}^{n-2}A$$
Where A is a non-singular matrix here
$$\left|AdjA\right|={\left|A\right|}^{n-1}$$
Where A is a non-singular matrix
A square matrix A is said to be a singular matrix if its determinant is zero, otherwise it is a non-singular matrix.
Inverse of a Square Matrix:
Let A be a square matrix of order n, then a square matrix B, such that $$AB=BA=I$$, is called inverse of A, denoted by $$A^{-1}$$
The formula for inverse matrix is given by
$$A^{-1}={\displaystyle \frac{1}{\left|A\right|}}\left(AdjA\right)$$
Properties of a inverse square matrix are given by
$$\left|A^{-1}\right|={\left|A\right|}^{-1}$$
$${\left(A^{-1}\right)}^{-1}=A$$
Now, from the given condition in the question we have
$$\Rightarrow {\left(AdjA\right)}^{-1}$$
Now, using the inverse of a square matrix formula mentioned above we get,
$$\Rightarrow {\left(AdjA\right)}^{-1}={\displaystyle \frac{1}{\left|AdjA\right|}}Adj\left(AdjA\right)$$
Now, using the properties of a adjoint matrix we can further convert them as
$$\Rightarrow {\left(AdjA\right)}^{-1}={\displaystyle \frac{1}{{\left|A\right|}^{n-1}}}{\left|A\right|}^{n-2}A$$
Now, on cancelling out the common terms we can further write it as
$$\Rightarrow {\left(AdjA\right)}^{-1}={\left|A\right|}^{-1}A$$
Now, let us further convert the left hand side accordingly using the properties of inverse matrix
$$\Rightarrow {\left(AdjA\right)}^{-1}={\left|A\right|}^{-1}{\left(A^{-1}\right)}^{-1}$$
Now, this can be further written using the properties as
$$\Rightarrow {\left(AdjA\right)}^{-1}=\left|A^{-1}\right|{\left(A^{-1}\right)}^{-1}[\because \left|A^{-1}\right|={\left|A\right|}^{-1}]$$
Now, from the formula of inverse of a square matrix this can be further written as
$$\therefore {\left(AdjA\right)}^{-1}=Adj\left(A^{-1}\right)$$
Hence, the correct option is (a).

Note: Instead of converting the value of the given function obtained by simplification we can also solve this question by finding the values of each option and then check with the value we got. Both the methods will give the same answer but it is a bit difficult to simplify all the options given.
It is important to note that we can convert the matrix value obtained accordingly using the respective properties without which we cannot directly get the result. It is also to be noted that considering the wrong property or neglecting any of the terms cannot give the result.