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Matrix Computation

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Here is the problem I need help with.

Let

A= -1 2
1 3

(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.

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Solution Preview

[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',

where
A = [a b
c d]

A' = [d -b
-c a]

|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
-1 -1]

A^-1 = (-1/5)*[3 -2
-1 -1]

A^-1 = [-3/5 2/5
1/5 1/5]

(b) A^3 = A*A*A

Matrix multiplcation:
C=A*B is defined as [c1,1 c1,2 = [a1,1 a1,2 * [b1,1 b1,2
...

Solution Summary

Matrix Computation is demonstrated.

$2.19
See Also This Related BrainMass Solution

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.

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