Week 9 Linear Algebra 2

Question 1

what do we do for homogenous compact matrix equations (Ax =v)

Answer 1

we set v and Ax to = 0

Question 2

the trivial solution to a homogenous equation is that x = 0 however when does this become unique

Answer 2

becomes the unique solution when detA ≠ 0

Question 3

what the condition in order for the homogenous equation to have an infinite number of non-trivial solutions

Answer 3

detA = 0

Question 4

give the general form of the characteristic equation for a 2x2 matrix

Answer 4

m^2 - m(A11 +A22) + A11A22 - A12A21 = 0

Question 5

what is the name for the solutions of the characteristic equation

Answer 5

eigenvalues

Question 6

what affects the number of eigenvalues

Answer 6

it is equal to the highest power of the polynomial characteristic equation

Question 7

how is the eigenvector found

Answer 7

found by solving (A - miI)ei = 0
where I is the identity matrix
mi is the eigenvalue
ei is the eigenvector

Question 8

what is true of the transpose of a symmetric matrix

Answer 8

A^T = A

Question 9

what is true of the transpose of an orthogonal matrix

Answer 9

A^T = A^-1

Question 10

define the trace of a matrix

Answer 10

trace of a matrix is equal to the sum of the elements on the leading diagonal

Question 11

how is the cross product of two matrices expressed

Answer 11

in terms of a determinant with leading row as the unit vectors i, j, k

Question 12

how is the scalar triple product of three matrices expressed a(b x c)

Answer 12

in terms of a determinant where the values of a are the leading row