Linear Algebra Answers

find all the values of of λ∈c such that λ(1+2i, 5+4i) =(3 +2i, 6 - i)

Good morning.

My question is:

Suppose v₁; v₂;...; v_m is linearly independent in V and w ∈ V .

Show that v₁; v₂; ...; v_m; w is linearly independent if and only if w ∉ span(v₁; v₂; :::; v_m).

Please assist.

Consider the given matrix B =   2 2 0 1 0 1 0 1 1   . Find detB and use it to determine whether or not B is invertible, and if so, find B −1 . (Hint: Use the matrix equation BX = I)

Consider the given matrix B =   2 2 0 1 0 1 0 1 1   . Find detB and use it to determine whether or not B is invertible, and if so, find B −1 . (Hint: Use the matrix equation BX = I)

Show that if A is an n × n matrix, then AAT

and A + A

T

are symmetric.

Show that if A is a matrix with a row of zeros (or a column of zeros), then A cannot have an inverse 

(6.1) Find det(C) if (1) C =  λ λ + 1 λ λ − 1  (6.2) Use the cofactor expansion to determine 2 0 0 0 3 1 2 0 2 −5 0 4 1 3 0 3 (6.3) Consider the matrix A =  1 4 2 3  (a) Compute A −1 (b) Find det(A **−1 ) (c) Deduce a relation (if it exists) between det(A) and det(A **−1 

Assume that A and B are matrices of the same size. Determine an expression for A if 2A − B = 5(A + 2B).

Consider the matrices A =   3 0 2 4 −6 3 −2 1 8   ,B =   −5 1 1 0 3 0 7 6 2   , C =   1 1 1 2 3 −1 3 −5 −7   , Verify the following expressions (where possible and give reasons) (i) A + (B + C) = (A + B) + C and A(BC) = (AB)C. (ii) (a − b)C = aC − bC and a(B − C) = aB − aC, where a = −2, b = 3 . (iii) (A T ) T = A and (A − B) T = A T − B T .

Let P(x) = x 2 − x − 6. Compute P(A) for A =  3 −1 0 −2