Matrix Algebra

Question 1

7! = ? x ? x ?!

Answer 1

7 x 6 x 5!

Question 3

How do you get the transpose of a matrix?

Answer 3

Rows become columns

Question 4

Answer 4

Question 5

What is an orthogonal matrix?

Answer 5

A square matrix whose transpose equals its inverse.

Question 6

What is the transformation matrix which reflects in the x-axis? (Or y=0)

Answer 6

1 00 -1

Question 7

What is the rotational matrix?

Answer 7

Cos (a) -Sin(a)Sin (a) Cos(a)where a is the anit-clockwise angle

Question 8

If R=rotation matrix, A = original coordinates and B = image coords after rotating:Write down the formula for calculating image coords?

Answer 8

B = R A

Question 9

If R=rotation matrix, A = original coordinates and B = image coords after rotating:Write down the formula for calculating original coords?

Answer 9

A = R^(-1) B

Question 10

What is the transformation matrix which reflects in the line y=x?

Answer 10

0 11 0

Question 11

What is the transformation matrix which reflects in the y-axis? (Or x=0)

Answer 11

-1 00 1

Question 12

What is the transformation matrix which increases (enlarges) the x & y coordinates by 3?

Answer 12

3 00 3

Question 13

What is the transformation matrix which reflects in the origin?

Answer 13

-1 00 -1

Question 14

What is the transformation matrix which decreases the x co-ordinate by half and increases the y co-ordinate by 4?

Answer 14

0.5 00 4

Question 15

For a rotational Matrix is it the clockwise or anti-clockwise angle you use?

Answer 15

Anti-clockwise

Question 16

What is A A^(-1) equal to?

Answer 16

I - the identitiy matrix

Question 17

What is I B = ?

Answer 17

B

Question 18

What are the conditions needed for A A^(T) = I

Answer 18

Matrix A must be a square orthogonal matrix

Question 19

What is the reflection using an angle matrix?

Answer 19

Cos (2a) Sin(2a)Sin (2a) -Cos(2a)where a is the angle made with the positve direction of the x-axis

Question 20

For the reflection matrix what is the condition for the angle used?

Answer 20

It is the angle measured from the positive direction of the x-axis: so lies between 0 and 180 or 0 and -180

Question 21

How do you find the transformation matrix if it has changed 2 sets of coordinates?

Answer 21

set up transformation matrix as:a bc dMultiply out both coordinate matrix seperatley to give 2 sets of eqns which can be solved using simlult eqns.

Question 22

How would you show that a matrix, A, has an inverse using matrix algebra?Ex. if A satisfys the equation 2A^(2) = A + I show that A is invertible

Answer 22

rearrange eqn to get:A(2A - 1) = Itherefore A has inverse as A x A^(-1) = I where the inverse matrix is (2A-1).

Question 23

How would you prove that a system of 3 eqns (3 planes or lines) have a solution (intersect at a point) without actually solving for that point?

Answer 23

Calculate the determinant and show that it is non-singular (not equal to zero)

Question 24

How would you prove that a 3x3 matrix can be inverted without actually inverting it?

Answer 24

Calculate the determinant and show that it is non-singular (not equal to zero)

Question 25

What is the det of A if A is an orthoganol matrix?

Answer 25

+1 or -1

Question 26

Does a matrix need to be square to have a transpose?

Answer 26

NO (Rows become colums)

Question 27

What is the inverse of the matrix A:a bc d

Answer 27

1/(det A) d -b -c a

Question 28

How do you get the inverse matrix of a 3x3 matrix A?

Answer 28

1. Set up augmented matrix with the identity matrix.2. Use ERO’s to move Identity matrix to left of augmented matrix.3. Right hand side of augmented matrix is inverse of A.

Question 29

Does AB = BA

Answer 29

In general - NO

Question 30

Does (AB)C = A(BC)

Answer 30

Yip

Question 31

(A’)’ = ?

Answer 31

A

Question 32

(A + B)’ = ?

Answer 32

A’ + B’

Question 33

(AB)’ = ?

Answer 33

B’A’

Question 34

(AB)^(−1) = ?

Answer 34

B^(−1)A^(−1)

Question 35

det (AB) = ?

Answer 35

det A det B

Question 36

A + B = ?

Answer 36

B + A