Linear Algebra Answers

A) Suppose A is a 3 x 3 matrix such that det (A) = 1/125 . Find det (5A^-1)


B)Suppose A is a 3 x 3 matrix such that det (A) = 5. Determine det [A^4]

A) Suppose A is a 3 x 3 matrix such that det (A) = 1/125 . Find det (5A^-1)


B)Suppose A is a 3 x 3 matrix such that det (A) = 5. Determine det [A^4]

Let the plane V be defined by ax + by + cz + d = 0 with at least one of a; b or c different from zero
and d >= 0.
Then the distance between V and the origin is d/(a^2+b^2+c^2)^1/2 : Prove this statement.

Explain the difference between a singular and a non-singular matrix. Show that a non-singular
matrix must be square.

Let A be a 3 x 3 matrix A with eigenvalues
1, –2, 2. Find the trace of A + A². Give
reasons to justify your answer.

Let W {(x, y, z) R3: x + y + z = 0}. Check
if W is a subspace of R3 . Find a non-zero
subspace U of R3 so that W intersection U = (0).

Check if (1, 3, 0) lies in the range of a linear
operator T on R3 given by
T (x1, x2 , x3) = (x1 , x2+ x3, x1– x2).
Is T one-one and onto ? Give reasons.

Suppose A and B are n x n matrices with A invertible. Prove that det ABA^-1 = det B

Let u = (4,2,-1), v = (3, 1, 1) and w = (0, 2, 1). Compute the following:
(i) 2v - 3w -u
(ii) u(w+v)
(iii) ||u . w ||
(iv) the orthogonal projection of u on w
(v) the vector component of u orthogonal to w

1. Verify if the vectors (3, 4, 5), (-3, 0, 5), (4, 4, 4), (3, 4, 0) are linearly independent.
2. Let A={b1,b2,b3} be a set of three-dimensional vectors in R3.
a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.
b. Prove that if the set A spans R3, then A is a basis of R3.