Check the statements if they are True or False: (a) Let A be a 2 times 2 real ma

Question

Check the statements if they are True or False: (a) Let A be a 2 times 2 real matrix which has a real eigenvalue. Then there is a 1 dimensional subspace, L in R^2 such that A upsilon stays inside L whenever upsilon is from L. (b) If {upsilon_1, upsilon_2, upsilon_3} are orthogonal vectors, then they are linearly independent. (c) The matrix A = [0 -1 1 0] has a real eigenvalue. (d) If A is a 3 times 3 real matrix, then it has a real eigenvalue. (e) ||c upsilon ||^2 = |c||| upsilon ||^2, for any vector upsilon and any scalar c.

Explanation / Answer

c.) false

A = 0 -1

1 0

to check eigen value ,

= 0-h -1

1 0-h

= (-h)(-h) - (-1)1

= h*h + 1

put this equals to zero ,

h*h + 1 =0

h*h = -1

h*h = i*i

h = +- i

b.) true because if it is a non zero set of orthognal vectors , then they are linearly indepedent .

d.) true because if any matrix A 3*3 is a real matrix ,then it has a real eigenvalue .

e.) ||cv||^2 = |c|||v||^2 false , here should be square of scalar c's absolute value .

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