Here is the problem I need help with.
A= -1 2
(a) Find A ^ -1
(b) find A ^ 3
(c) Find (A ^ -1) ^ 3
(d) (The part that really has me stumped) Use your answers to (b) and (c) to show that (a ^ -1) ^ 3 is the inverse of A ^ 3.
[Sorry for any lack of clarity due to the text box medium, hopefully this is clear enough].
(a) The inverse matrix is defined such that A*A^-1 = I, where I is the identity matrix.
The formula for determining the inverse of a matrix is A^-1 = (1/|A|)*A',
A = [a b
A' = [d -b
|A| = ad - bc (the determinant)
So, in this case a=-1, b=2, c=1, d=3. When you plug that into the formula you get:
A^-1 = (1/(-1*3-2*1))*[3 -2
A^-1 = (-1/5)*[3 -2
A^-1 = [-3/5 2/5
(b) A^3 = A*A*A
C=A*B is defined as [c1,1 c1,2 = [a1,1 a1,2 * [b1,1 b1,2
Matrix Computation is demonstrated.
Computing a Proof Regarding Eigenvectors and Matrices
D and E are nxn matrices, E is invertible, DE = ED, and u is an eigenvector for D corresponding to x=5.
a. Show that Eu is also an eigenvector for D corresponding to x=5.
b. Show that u is an eigenvector for D^2.
c. Show that u is an eigenvector for
D^2 - 3D.