MATH 308L - FIRST WORKSHEET

(1) Samurai Jack is my favorite cartoon character. He is extremely knowl- edgeable and wise. He travelled through time to see me “yesterday”. He asked me to test his knowledge on row reduced echelon forms of matri- ces(?!?). So I did. I did not go easy on him. I asked him the following questions.

(i) Suppose you have two non-singular matrices A and B. What is the relationship between the two matrices - • RREF (A) · RREF (B) • RREF (AB)? Are they always equal ? Are there examples where they are not equal ? (ii) Then, I changed the hypothesis so that A and B are both singular matrices. I proceeded to ask him the same questions as above. (iii) Now let’s change the hypothesis again. Suppose A is a singular matrix and B is a non-singular matrix. Same questions !

Jack was stumped at all three questions (He is still my favorite cartoon character though). Can you do better at these questions than Samurai Jack ?

(2) A few hours later, Aku (the evil character in the cartoon show) also travelled through time to see me “yesterday”. Aku is less knowledgeable and so I had to teach him a few things about matrix multiplication through some exercises.  1 0  (a) Let D = . Find D2,D3,D4 etc. Do you observe a 0 −10 pattern ? (b) Let A be an invertible 2 × 2 matrix. Find a simplified expression for (ADA−1)n for any positive integer n.  3 4   1 0  (c) Let M = and let B = . Can you find B4, by 1 1 0 −10 using the matrices D and M, and without explicitly computing B4.

1  2 1 3  (3) Let A and B be two 3 × 3 matrices. The matrix A =  0 1 0  is 0 0 1 invertible. Also, we have the following relation.

 1 1 1  −1 ABA =  0 0 0  0 0 1

Solve for B. (Be careful. Matrix multiplication is not commutative. So, you cannot say things like “ABA−1 = AA−1B = B”. This is NOT true).  1 3  (4) Let A = . The goal of this question is to find values of the 1 −1 scalar k such that the following vector equation

A~x = k~x.

has a non-trivial solution. Now, this should scare you. We are trying to solve for ~x and the scalar k at the same time ! However, fear not.

(a) Suppose you have found a scalar k such that the vector equation A~x = k~x has a non-trivial solution. What can you say about the matrix A − kI2 ?

(b) Let x1 and x2 be two variables. Can you find values of k such that following system of equations

(1 − k)x1 + 3x2 = 0,

1x1 + (−1 − k)x2 = 0,

has infinitely many solutions. (c) Now for each value of k obtained in the previous part, solve the system of equations given earlier. (d) Find the values of the scalar k such that the vector equation A~x = k~x has a non-trivial solution. For each such scalar k, list the non- trivial solutions for the vector ~x.

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