Homework Statement
Let A and B be diagonalizable 2 x 2 matrices. If every eigenvector of A is an eigenvector of B show that AB = BA.
Homework Equations
D = PA(P^-1)
The Attempt at a Solution
Since the eigenvectors are equivalent, wouldn't it hold true that P_A = P_B?
If I have...
Homework Statement
Let A be an invertible 3x3 matrix. Suppose it is known that:
A =
[u v w
3 3 -2
x y z]
and that adj(A) =
[a 3 b
-1 1 2
c -2 d]
Find det(A)
(answer without any unknown variables)
Homework Equations
The Attempt at a Solution
I found A^(-1) to be equal...
Homework Statement
Let T1 be the reflection about the line 2x–5y=0 and T2 be the reflection about the line –4x+3y=0 in the euclidean plane.
(i) The standard matrix of T1 o T2 is: ?
Thus T1 o T2 is a counterclockwise rotation about the origin by an angle of _ radians?
(ii) The...
Homework Statement
Find points P,Q which are closest possible with P lying on line:
x=7-5t, y=-5+11t, z=-3-1t
and Q lying on line:
x=-354-8t, y=-194+12t, z=-73+7t
*the line joining P + Q is perpendicular to the two given lines.
Homework Equations
Projection formula, cross...
Homework Statement
1. Let A =
[-8 k
0 -8]
Then A is diagonalizable exactly for the following values of k
2. Let B =
[-8 k
0 1]
Then B is diagonalizable exactly for the following values of k
Homework Equations
-Equations for eigenvalues, eigenvectors... and D=PA(P^-1)
-A...
Homework Statement
Determine the distance between the parallel planes –4x–4y+1z=–1 and 8x+8y–2z=12
Homework Equations
Proj_n_v = ((vn)/(nn))n
The Attempt at a Solution
I thought I understood how to do this, but I am not getting a correct answer for it. What I did was:
I made the...
Homework Statement
Let A=
[-1 4
3 -12]
Find two 2x2 matrices B and C such that AB=AC but B does not equal C .
Homework Equations
The Attempt at a Solution
I was going through my book, and am a bit confused with this problem. How would I solve this?
I know it's easy to prove...
Homework Statement
Suppose that the 2x2 matrix A has eigenvalues lambda = 1,3 with corresponding eigenvectors [2,-1]^T and [3,2]^T. Find a formula for the entries of A^n for any integer n. And then, find A and A^-1 from your formula.
Homework Equations
Ax = lambda X
(P^-1)AP = D
A =...
Homework Statement
1. Give an example of a 2x2 real matrix A such that A^2 = -I
2. Prove that there is no real 3x3 matrix A with A^2 = -I
Homework Equations
I think these equations would apply here?
det(A^x) = (detA)^x
det(kA) = (k^n)detA (A being an nxn matrix)
det(I) = 1
The...
Homework Statement
Prove that
[1 a b
-a 1 c
-b -c 1]
is invertible for any real numbers a,b,c
Homework Equations
A is invertible if and only if det[A] does not equal 0.
The Attempt at a Solution
I'm not sure if I'm going about this in the correct way;
Would I prove this...
I'm a bit confused how I would solve these problems of matrix inversion. I know the basic properties of inversion, but need some explanation on how to prove or disprove questions such as the ones below: I need to justify each these as true or false.
1. If A is an invertible, real matrix... is...