Linear Algebra Answers

Let.

1             0             0

0             3             6

0             -1           -2

Verify that C ² = C holds. Find the eigenvalues and eigenvectors of C. 

Let. (Matrix)

2             0             1

2             -2           2

0             4             1

(a) Find ^A2 and ^A3 , and verify that: ^{A 3} − ^{A 2} − 12A = −12i

holds, where I stands for the identity matrix.

(b) Find ^A−1 by multiplying the equation above on both sides by ^A−1

Compute the determinant, and use Gauss-Jordan elimination to find the inverse of the following matrix (if it exists).

218        0             0

0             218        2

0             1             1

Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.

Verify Rank-nullity theorem for the linear transformation T:R³----->R³ defined by T(x,y,z)=(x-y, 2y+z, x+y+z).

4x-3y+z=-8

-2x+y-3z=-4

x-y+2z=3

Use Sylvester's theorem to show that e^A=e^x[cos hx. Sin hx ]. Where,

[ Sine hx cos hx]

A= [ x. x]. (CO5)

[x. x]

Investigate the linear dependence and independence of the vector X1=(1,2,3), X2=(2,-1,3), X3=(0,1,2), X4=(-3,7,2)

Use matrix method to solve d^2x/dx^2+4x+2=0, x(0)=1 , x(0))=0

R^3 is a inner product space over the inner product

<(x1,x2,X3),(y1,y2,y3)> = x1y1+ x2y2 - x3y3

True or false with full explanation