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
Does a matrix need to be square to have a transpose?
NO (Rows become colums)
What is the inverse of the matrix A:a bc d
1/(det A) d -b -c a
How do you get the inverse matrix of a 3x3 matrix A?
- 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.
Does AB = BA
In general - NO
Does (AB)C = A(BC)
Yip
(A’)’ = ?
A
(A + B)’ = ?
A’ + B’
(AB)’ = ?
B’A’
(AB)^(−1) = ?
B^(−1)A^(−1)
det (AB) = ?
det A det B
A + B = ?
B + A
