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1. 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...
2. 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...
3. 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...
4. 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...
5. 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...
6. 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...
7. 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...
8. 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 =...
9. 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...
10. 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...
11. Showing that a matrix is invertible

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...