# Recently Submitted Homework Problems: linear algebra

527. How to find the vector equation of the line through A(1,2,-1) & perpendicular to the lines X=(1,-1,2)+ r(2,2,3) and X= (1,-1,2)+ r(1,-1,0)? (Answered)1211. Writing Proof: Let A be an m x n-matrix. Prove that rank A = m then there exists a matrix B such that AB = Im. Similarly, if rank A = n then there exists a matrix C such that CA = In.

*m in Im is subscript for I & same for n in In(Answered)

1215. Find all n x n matrices A such that Q^(-1)AQ=A for all invertible n x n-matrices Q.

What I worked out: If Q is invertible, then Q^(-1)AQ = A -->Q(Q^(-1)AQ) = QA-->(QQ^(-1))(AQ) = QA --> InAQ = AQ = QA

So, all n x n matrices A must commute with all n x n invertible matrices Q.

(Answered)

1218. Find all n x n matrices A such that Q^(-1)AQ=A for all invertible n x n-matrices Q. What I worked out: If Q is invertible, then Q^(-1)AQ = A -->Q(Q^(-1)AQ) = QA-->(QQ^(-1))(AQ) = QA --> InAQ = AQ = QA So, all n x n matrices A must commute with all n x n invertible matrices Q.(Answered)

3261. Determine whether the subset S of M2x2 is a subspace of the set of all upper triangular matrices.(Answered)

3700. A triangle has vertices with coordinates (-4,6) (2,-6) and (8,10)

Write the equations of the lines containing each of the sides of the triangle.(Answered)

3700. A triangle has vertices with coordinates (-4,6) (2,-6) and (8,10)

Write the equations of the lines containing each of the sides of the triangle.(Answered)

4664. Find the eigenvalues and corresponding eigenvectors of the following matrix:

0 3 0

3 -3 0

0 -3 6

(Answered)

5050. Show that any spanning set of a (finite-dimensional) vector space V can be cut down to a basis for V(Answered)