Need help to proof Adjugate Matrix

[nasromeo]

Alo! I kinda need some assistance to proof this:

"Show that adj(adj A) = |A|^(n-2). A, if A is a (n x n) square matrix and |A| is not equal to zero"

NOTE: 1) adj(A) = adjugate of matrix A,
2) |A| = determinant of A,
3) ^ = power

I've tried to work around the equation using the formula: A^-1 = |A|^-1. adj(A), BUT doesn't seem to work at all. Sooo HELP!..and thanks in advance .

 

[matt grime]

The definition of Adj(A) is *not* given by a formula involving A inverse. It is defined even if A is not invertible.

Adj(X) satisfies X*Adj(X)=det(X)I.

 

[red_dog]

We have $$A\cdot A^*=\det A\cdot I_n$$ (1)
Then $$det A\cdot\det A^*=(\det A)^n\Rightarrow\det A^*=(\det A)^{n-1}$$
Applying (1) for $$A^*$$ we have
$$A^*\cdot (A^*)^*=\det A^*\cdot I_n$$.
Multiply both members by $$A$$
$$A\cdot A^*(A^*)^*=det A^*\cdot A\Rightarrow \det A\cdot (A^*)^*=(\det A)^{n-1}\cdot A\Rightarrow$$
$$\Rightarrow (A^*)^*=(det A)^{n-2}\cdot A$$