Matrices

Question 1

what does it mean if matrix multiplication is associative, using A B and C as examples

Answer 1

• (AB)C = A(BC) = ABC

- or A(B+C) = AB + AC

Question 2

what does it mean if matrix multiplication is not commutative, using A and B as examples

Answer 2

AB does not = BA

Question 3

what does the transpose of a transpose of a matrix equal

Answer 3

the original matrix

Question 4

what are the 2 things ((AB)^T)ij can equate to

Answer 4

• (AB)ji (notice i and j swapped)

- (B^T * A^T)ij ( notice i and j did not swap but B and A did)

Question 5

what does a . b equal in terms of a^T

Answer 5

a . b = a^T * b

Question 6

what is the condition in order for the inverse of a matrix to not exist

Answer 6

the determinant of the matrix must be 0

Question 7

how does multiplying a matrix by a constant c change the determinant

Answer 7

it also multiplies the determinant by c

Question 8

what is an easy way to spot whether the det of a matrix is 0

Answer 8

if 2 rows or columns are equal

Question 9

does the det(A) = det(A^T)

Answer 9

yes

Question 10

what is the det(AB) also equal to

Answer 10

det(BA)

Question 11

what is the 2x2 matrix for rotations (all matrix notations are read downwards first

Answer 11

• cosθ, sinθ

- -sinθ, cosθ

Question 12

what direction does the inverse of a regular rotation matrix rotate a vector and why

Answer 12

• in the clockwise direction

- because all the θs turn into -θs

Question 13

what is the inverse of a rotation matrix equal to

Answer 13

its transpose

Question 14

what is the matrix for rotation about the x-axis

Answer 14

• 1 0 0
• 0 cosθ sinθ
• 0 -sinθ cosθ

Question 15

what is the matrix for the rotation about the y-axis

Answer 15

• cosθ 0 -sinθ
• 0 1 0
• sinθ 0 cosθ

Question 16

what is the matrix for the rotation about the z-axis

Answer 16

• cosθ sinθ 0
• -sinθ cosθ 0
• 0 0 1

Question 17

are rotations in 2D commutative (does doing A then B = doing B then A)

Answer 17

yes

Question 18

what is the only scenario in 3D where rotations are commutative

Answer 18

if its about the same axis

Question 19

for a rotation matrix Q, what does QQ^T or QQ^-1 equal

Answer 19

the identity matrix I

Question 20

simply speaking, how would you prove that a matrix is a rotation matrix

Answer 20

by showing that it doesnt change the length of the rotated vector

Question 21

what is the proof showing that the length of a rotated vector x doesnt change ( starting from x’ = Qx)

Answer 21

• x’ . x’ = (x’)^T * x’ (length of new vector squared)
• (x’)^T * x’ = (Qx)^T * Qx
• = x^T * Q^T * Q * x (you can expand brackets if you flip the order)
• as Q^T*Q = I, above = x^T * x

Question 22

what is the difference between the determinant of a rotation and reflection matrix

Answer 22

• the det of a rotation matrix is 1

- the det of a reflection matrix is -1

Question 23

if R = Q^T, where R rotates the coordinate system by an angle and Q^T is the transpose of a rotation matrix that would normally rotate a vector by and angle, how would you present the change of a coordinate system in 3D using ‘a’ as the vector

Answer 23

a’ = Ra

Question 24

if a’ = Ra, what is the relationship between a’ and a when R is turned to R^-1

Answer 24

a = (R^-1)a’

Question 25

if A = a linear transpose in a coordinate system, what is the formula showing the relationship between A and the matrix R

Answer 25

A’ = RAR^T