Matrices

Question 1

in terms of column vectors, Ax=b becomes

Answer 1

x1v1+x2v2+…+xnvn=b

Question 2

for a 2x2 matrix, A, det(A) is

Answer 2

A11A22 - A12A21

Question 3

for an n by n matrix, we calculate the determinant using

Answer 3

the laplace expansion

Question 4

laplace expansion: what is j

Answer 4

any value between 1 and n

Question 5

laplace expansion: what is Cij

Answer 5

the cofactor of matrix element Aij

Question 6

laplace expansion: what is Mij

Answer 6

the minor
(det of A after row i and column j are removed)

Question 7

det(A^T)=

Answer 7

det(A)

Question 8

det(lambda A)=

Answer 8

lambda^n det(A) for an nxn matrix

Question 9

det(AB)=

Answer 9

det(BA)=det(A)det(B)

Question 10

effect on determinant when two rows/columns are interchanged

Answer 10

changes sign of det

Question 11

if two rows/columns are multiples of each other then det =

Answer 11

0

Question 12

multiplying a row by a non-zero scalar does what to det?

Answer 12

multiplies by same scalar

Question 13

a square matrix is singular if

Answer 13

det(A)=0

Question 14

a non-singular square matrix A has inverse A^-1, defined by

Answer 14

A^-1A = AA^-1 = I

where I is the identity matrix

Question 15

inverse can be calculated by

Answer 15

A^-1=C^T/det(A)

Question 16

(A^-1)^-1=

Answer 16

A

Question 17

inverse of the transpose of A =

Answer 17

transpose of the inverse of A

Question 18

(AB)^-1=

Answer 18

B^-1A^-1

DOES NOT EQUAL A^-1B^-1

Question 19

det(A^-1(=

Answer 19

1/det(A)

Question 20

the hermitian conjugate of a matrix is found by taking

Answer 20

the complex conjugate and the transpose

Question 21

a matrix is hermitian if

Answer 21

it is equal to its hermitian conjugate

Question 22

diagonal matrix

Answer 22

all non-diagonal elements are zero

Question 24

upper triangular matrix

Answer 24

if Aij=0 for i>j

Question 25

lower triangular matrix

Answer 25

if Aij=0 for i<j

Question 26

symmetric matrix

Answer 26

equal to its transpose

Question 27

normal matrix

Answer 27

if A^crossA=AA^cross

hermitian matrices are examples of these

Question 28

orthogonal matrix

Answer 28

transpose is equal to the inverse

Question 29

unitary matrix

Answer 29

A^cross = inverse

equivalent of orthogonal matrices for complex matrices

Question 30

if for a non-zero vector, Ax=lambda x then

Answer 30

x is an eigenvector of A and lambda is the corresponding eigenvalue

Question 31

any scalar multiple mu x of x is also

Answer 31

an eigenvector with the same eigenvalue

Question 32

due to scalar multiple, it is more convenient to use

Answer 32

normalised eigenvectors (abs value of 1)

Question 33

steps to calculate eigenvalues and eigenvectors

Answer 33

1. rewrite eigenvalue equation in form (A-lambda I)x=0
2. find characteristic equation det(A-lambda I)=0
3. roots are eigenvalues
4. sub eigenvalues back into first eqn for eigenvectors
5. normalise eigenvectors if required