Note: This property holds for square matrices.
If, we are given matrix A then
How to prove that
? where adj(A) is adjoint of A and det(A) is determinant of A.
We know that inverse of matrix is calculated using formula:
Multiplying this equation by A, we can write as
and
and
From above, we can say that det(A)I=A.adj(A) and det(A)I=adj(A).A
From above equations, we can say that A.adj(A)=adj(A).A=det(A)I
which is the desired result.
Deepak Gautam says
The above proof is done assuming A is invertible, that A(inverse) exists. What if A(inverse) doesn’t exists. What if A is not invertible. How will the proof begin then?
Slim Shady says
A^(-1) doesn’t exist if and only if det(A) = 0. That is, the theorem above is generalized for both invertible and non-invertible matrices.