Assignment 4
I saw several HW papers in which points were taken off for finding the the leading columns of A in problem 2, even though the answer given was correct. Just to make sure that everyone knows how to do this, I'm giving the full, correct answer to the problem.
Consider the matrix A =
1 −3 2 −2 2 −1 3 −2 1 −3 2 −6 5 −3 5Find the reduced echelon form of A, the rank of A, the nullity of A, and the leading columns of A. Determine whether the columns of A are LI or LD. Solve Ax = 0.
Here is the answer.
rref(A) =
1 -3 0 0 4
0 0 1 0 0
0 0 0 1 1
rank(A) = 3, nullity(A) = 2. Columns of A are LD, because rank(A) < 5.
Leading columns of A are #1, #3, and #4 in A (NOT in rref(A)).
1 2 −2
−1 −2 1
2 5 −3
The solution to Ax = 0 is
Assignment 5
Some people were confused about the answer to problem 3, in which you were asked to invert A below. Here is the answer.
A =
1 -1 4 2
-2 1 0 1
0 0 4 3
1 -1 5 3
A− 1 =
-1 -1 1 0
2 -1 3 -4
3 0 1 -3
-4 0 -1 4