Linear Algebra Answers

If a polynomial p(x) divided by x-2 then remainder is one and when it is divided by x+1 the remainder is -2 find the remainder when the given polynomial divided by (x-2)(x+1)

 Let A be any n x n matrix and let P be an n x n orthogonal matrix. Prove that jAj D jP

APj:

Suppose a is element of R. Show that the set of continuous real-valued functions f on the interval [0;1] such that integral f from 0 to 1 = a is a subspace of R[0;1] if and only if a = 0

Give an example of 2 × 2 matrix with non-zero entries that has no inverse

Find an expression for a square matrix A satisfying A 2 = In, where In is the n × n identity matrix

Use Cramer’s rule to solve for y without solving for x, z and w in the system 2w + x + y + z = 3 −8w − 7x − 3y + 5z = −3 w + 4x + y + z = 6 w + 3x + 7y − z = 1 

Prove that if A is a square matrix then AAT and A + A T are symmetric.

Suppose v₁v₂..v_mis linearly independent in V and w element of V. Show that v_1,v_2,...v_m;w is linearly independent if and only if w = not element of span(v_1,v_2,...v_m).

Let P(x) = x2 - x - 6. Compute P(A) for matrix A = [3, -1and 0 -2]

(6.1) Find the values of a, b and c such the matrix below is skew symmetric.

0 0 d

0 2a − 3b + c 3a − 5b + 5c

2 0 5a − 8b + 6c

(6.2) Give an example of a skew symmetric matrix.

(6.3) Prove that A² is symmetric whenever A is skewsymmetric.

(6.4) Determine an expression for det(A) in terms of det(A^T) if A is a square skewsymmetric.

(6.5) Assume that A has an odd number of rows and also an odd number of columns. In this particular case, show that det(·) is an odd function.