Matrix Algebra Flashcards
7! = ? x ? x ?!
7 x 6 x 5!
How do you get the transpose of a matrix?
Rows become columns
What is an orthogonal matrix?
A square matrix whose transpose equals its inverse.
What is the transformation matrix which reflects in the x-axis? (Or y=0)
1 00 -1
What is the rotational matrix?
Cos (a) -Sin(a)Sin (a) Cos(a)where a is the anit-clockwise angle
If R=rotation matrix, A = original coordinates and B = image coords after rotating:Write down the formula for calculating image coords?
B = R A
If R=rotation matrix, A = original coordinates and B = image coords after rotating:Write down the formula for calculating original coords?
A = R^(-1) B
What is the transformation matrix which reflects in the line y=x?
0 11 0
What is the transformation matrix which reflects in the y-axis? (Or x=0)
-1 00 1
What is the transformation matrix which increases (enlarges) the x & y coordinates by 3?
3 00 3
What is the transformation matrix which reflects in the origin?
-1 00 -1
What is the transformation matrix which decreases the x co-ordinate by half and increases the y co-ordinate by 4?
0.5 00 4
For a rotational Matrix is it the clockwise or anti-clockwise angle you use?
Anti-clockwise
What is A A^(-1) equal to?
I - the identitiy matrix
What is I B = ?
B
What are the conditions needed for A A^(T) = I
Matrix A must be a square orthogonal matrix
What is the reflection using an angle matrix?
Cos (2a) Sin(2a)Sin (2a) -Cos(2a)where a is the angle made with the positve direction of the x-axis
For the reflection matrix what is the condition for the angle used?
It is the angle measured from the positive direction of the x-axis: so lies between 0 and 180 or 0 and -180
How do you find the transformation matrix if it has changed 2 sets of coordinates?
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.
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
rearrange eqn to get:A(2A - 1) = Itherefore A has inverse as A x A^(-1) = I where the inverse matrix is (2A-1).
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?
Calculate the determinant and show that it is non-singular (not equal to zero)
How would you prove that a 3x3 matrix can be inverted without actually inverting it?
Calculate the determinant and show that it is non-singular (not equal to zero)
What is the det of A if A is an orthoganol matrix?
+1 or -1
