Matrices

Question 1

In matrix multiplication does AB = BA

Answer 1

No

Question 2

Transpose

Answer 2

First row becomes the first column, second row becomes the second column and so on…

Question 3

Scheme for square matrices

Answer 3

(A|In) to (C|D) where C is in row echelon form

Question 4

Invertibility

Answer 4

AB = In = BA, A is the matrix, B is the inverse of A and when multiplied they leave the identity matrix

Question 5

If A is investable what does that mean about the determinant

Answer 5

det(A) ≠ 0

Question 6

If the determinant of A is non zero how can you find the inverse of A

Answer 6

1/det(A) multiplied by adj(A), the adjacent of A is when the leading diagonal swap and the remaining diagonal swap signs

Question 7

Summary theorem (if and only if)

Answer 7

1. A is invertible
2. Ax = b has a unique solution
3. Ax = 0
4. (A|In) to (In|B)
5. det(A) ≠ 0

Question 8

Diagonalisation theorem

Answer 8

(P inverse)AP = D for some diagonal D if and only if A has n linearly independent eigenvectors

Question 9

Symmetric matrices

Answer 9

Symmetrical across the leading diagonal