Matrices

Question 1

When is a matrix homogenous?

Answer 1

When in Ax=b is =0

Question 2

A matrix is inconsistent when…

Answer 2

There are no solutions

Question 3

Linearly independent for a set of vectors means…

Answer 3

There is only a trivial solution for c1v1 + c2v2 + c3v3 = 0

Question 4

A set of vectors is linearly dependent if…

Answer 4

The all live on the same line or plane

Question 5

Rank

Answer 5

Maximal number of linearly independent row vectors

Question 6

What is row echelon form / gaussian elimination?

Answer 6

Interchanging / adding / negating rows. Aim for zeros at bottom and leading entries at right

Question 7

Rouché-Capelli to find if a matrix has solutions

Answer 7

rank [A] = rank [A|b]✅
(∞ if rank < unknowns)
rank [A] < rank [A|b] ❌

Question 8

Determinant of a 2x2 matrix [a b, c d]

Answer 8

|A| = ad − bc

Question 9

Determinant of 3x3 matrix

Answer 9

Top values times by their minors, then +-+

Question 10

Eiegenvalue equation

Answer 10

det(A-λI)=0
λ into original equation for eigenvector
Normalised vector by inputting unit

Question 11

Trace of a matrix

Answer 11

Adding the values on the leading diagonal

Also = λ1 + λ2 + λ3

Question 12

λ1 x λ2 x λ3 =?

Answer 12

Determinant

Question 13

Are eigenvectors from different eigenvalues always linearly independent?

Answer 13

Yes

Question 14

Is it true that a square matrix over a complex field has at least on eigenvalue?

Answer 14

Yes

Question 15

Λ

Answer 15

Matrix of eigenvalues

Question 16

Q modal matrix

Answer 16

Matrix of eigenvectors

Question 17

Similarity transformation - used in diagonalization

Answer 17

Λ = Q⁻¹AQ

Question 18

Putting a matrix to a power

Answer 18

Use AQ = QΛ

Aⁿ = QΛⁿQ⁻¹

Question 19

Solving first order equations with eigenvalues

Answer 19

Ẋ = AX = QΛQ⁻¹ X
Q⁻¹Ẋ = ΛQ⁻¹ X
To Ẏ = ΛY
Finally X = QY
Use boundary conditions for unknowns

Question 20

Repeated eigenvalues (multiplicity)

Answer 20

Produce an equal number of eigenvectors by setting one parameter to zero, giving linearly independent vectors. If cannot => non diagonisable

Question 21

Symmetry in matrices

Answer 21

A = A^T transpose

Question 22

What does symmetry in a matrix tell you?

Answer 22

For a real matrix - eigenvalues are all real

Can always be diagonalised