Linear Algebra Answers

Let A = (a ij) n x n, where n >= 3, such that a ij = i - j for each 1 <= i <= n and 1 <= j <= n. Prove that det(A) = 0

Let a 4x4 matrix M =
7 5 -1 8 //
7 5 -2 3 //
7 5 5 9 //
7 5 6 10

Find a 3 x 3 matrix A of the form
3 -1 -5
3 * *
3 * *

such that det(M) = 7 det(A)

Let A be an m x r matrix, B be an m x s matrix, C be an r x n matrix, and D be an s x n matrix.
Prove that
(A B) where A and B are side by side in the 1 x 2 bracket, multiply (C D) where C is on top of D in the 2x1 bracket = AC + BD

Give examples of 3 X 3 matrices A,B, and C satisfying the following conditions: A^2 = 2A, B^2 = 2B, C^2 =/= 2C, det(A) = 0, det(B) = det(C) = 8.

Let A be a 3 x 3 matrix such that A^2 = 2A. Prove that det(A) must be either 0 or 8.

For the vector space P2,find the dual basis of {1+x,1+2x,1+x+x2}?

Let A be an m x r matrix, B be an m x s matrix, C be an r x n matrix, and D be an s x n matrix. Prove that
(A B) (C
D) -> C on top of D
= AC + BD

Check that B={1,x+1,(x+1)(x+1)} is a basis for p2.
Find the co-ordinates of 2+x+3x2 with respect to this basis.

Consider the basis S = {v1, v2, v3} for R3, where v1 = (1, 2, 1), v2 = (2, 9, 0), and v3 = (3, 3, 4), and let T:R3 rightarrow R2 be the linear operator such that T(V1) = (1, 0), T(v2) = (- 1, 1), T(v3) = (0, 1) Find a formula for T(x1, x2, x3), and use that formula to find