# Search results

1. ### Diagonalizable matrices

Thanks HallsofIvy! I appreciate the explanation and for assuring me of that proof!!
2. ### Diagonalizable matrices

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...
3. ### Inverse Matrix Question

Thanks again Dick for clearing this up for me!
4. ### Inverse Matrix Question

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...
5. ### Composite Matrix Transformation - Reflection

Hi, Thanks again HallsofIvy. I used the unit vectors in my formula, and it seems to come out with the same answer; except I tried the technique you gave and I still come up with [(-7/25) (24/25) (24/25) (7/25)] for the second Matrix. I guess I'm making some calculation error, as...
6. ### Closest possible points on skew lines

Thanks for the reply Dick! Ahh k, got it now!! )
7. ### Composite Matrix Transformation - Reflection

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...
8. ### Closest possible points on skew lines

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...
9. ### Matrix Diagonalizable Question

Ahhh, Okay. I don't know why I didn't see that before... obviously then, there are two parameters for the 0 matrix, which would make it invertible... the parameters there would be t[1,0] and s[0,1] (or as you said any vector <x, y>. And thanks Dick, I see that when you have a k value other than...
10. ### Matrix Diagonalizable Question

Hi Dick, Thanks a lot for the help! The question was only asking for which k values, so your explanation of k=0 for the first one solved my mistake. But, if you could just clear something up for me that I don't really understand: As you mention the first one is diagonalizable if k=0, to...
11. ### Matrix Diagonalizable Question

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...
12. ### Distance between two parallel lines

Aa! Can't believe I made such a stupid mistake. Thanks!! So, the answer would just be (5/33)(sqrt33), correct?... As I would keep P as (0,0,-1), but make A (0,0,-6); and get the vector AP = (0,0,5) So, projecting AP onto the normal, and then getting the distance of the projection would...
13. ### Linear Algebra; AB=AC

Great, thanks guys! I set each to equal the zero matrix; so ended up getting a = 4c, and b = 4d. I used arbitrary numbers for C and D, and got two separate matrices B and C such that AB = AC... for example, these worked out: [4 8 1 2] and [12 16 3 4] Thanks for clearing that up for me!
14. ### Distance between two parallel lines

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...
15. ### Linear Algebra; AB=AC

Thanks for the help! This may sound like a stupid question, as I've encountered this problem before in this course... how would I solve for 4 unknowns in this way; I haven't taken Math for 4 years before Linear Algebra, so I'm rusty on how to do these types of equations. I would set all...
16. ### Linear Algebra; AB=AC

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...
17. ### Strange question regarding eigenvectors / eigenvalues

Ahhh, ok. Thanks for clearing that up for me; understand it now. Thanks!
18. ### Strange question regarding eigenvectors / eigenvalues

Hi Thanks. Just for clarification though, why would I not say A^(n) = (PDP^(-1))^n Why is it just D that I raise to the n, and not the whole right of the equation?
19. ### Strange question regarding eigenvectors / eigenvalues

Hi Dick. Thanks for your explanation! So, my formula would be: A^(n) = P(D^n)P^(-1) P is obviously just [2 3 -1 2] and P^(-1) would then just be its inverse, or [(2/7) (-3/7) (-1/7) (2/7)] Then you just use this P, P^(-1) and the given D^n to get the answer of A^n. So, then...
20. ### Strange question regarding eigenvectors / eigenvalues

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 =...
21. ### Linear Algebra - Determinant Properties

Ok got it. Thanks for the help!
22. ### Linear Algebra - Determinant Properties

Okay, so I could use properties from linear transformations and say: [0 -1 -1 0] = -I, which equals: [cos180 -sin180 sin180 cos180] Then, since A^2 = [cos180 -sin180 sin180 cos180] I need the square root of this, which is: [cos90 -sin90 sin90 cos90] So then A = [0 -1 1 0]...
23. ### Linear Algebra - Determinant Properties

Yeah, thanks! Oops! Well, since I am looking for an example of a matrix A where A^2 = -I, I could use the example A = [0 -1 1 0] Since this squared = -I But, I found this using trial and error. Is there any other way to answer this question...? It seems the question is too easy if it's...
24. ### Linear Algebra - Determinant Properties

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...

Thanks!
26. ### Linear Algebra - invertible matrix; determinants

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...
27. ### Showing that a matrix is invertible

Thanks Mark! It's very clear now. I appreciate it tons!
28. ### Showing that a matrix is invertible

Hey Mark Thanks for that explanation; it helped a great deal! I understand how to prove that a matrix is not invertible; to prove that AX=0 has exactly 1 solution etc. In regards to that last question: if a matrix B satisfies: (B^3)-2B+I=0, then the matrix (B-I) cannot be invertible? Although...
29. ### Showing that a matrix is invertible

Thanks Mark, This makes a lot more sense to me now. So on other problems that I may have for proving invertibility of arbitrary matrices, I should look for a case where the nxn matrix = 0 (let's say matrix A). If matrix A can be = to 0 (the 0 matrix), then it cannot have an A^-1, and therefore...
30. ### Showing that a matrix is invertible

Thanks again. That makes much sense. The only other thing is, when I find what is equal to 0, why is it that this means there would be no inverse? As you explained before, is it just because the matrices that carry nonzero vectors to zero are non-invertible? So if I use the concept of...