Linear Algebra Answers

Let "A = \\begin{pmatrix}\n 2 & 1 \\\\\n 5 & 3\n\\end{pmatrix}", the A^-1 = 1/p "\\begin{pmatrix}\n m & n \\\\\n s & t\n\\end{pmatrix}"

What is the value of p, n, s, t.

Let "A = \\begin{pmatrix}\n 1 & -2 & 4 \\\\\n 2 & -4 & 8 \\\\\n -1 & 0 & -1\n\\end{pmatrix}"

The matrix A has an inverse. True or false? Provide a reason/show your working.

Let "A = \\begin{pmatrix}\n 1 & 0 & 3\\\\\n 0 & 4 & 5 \\\\\n 1 & 2 & 6\n\\end{pmatrix}"

What is the contactor of the entry A₂₃ = 5

1. 2
2. -2
3. 10
4. -10

Let "B = \\begin{pmatrix}\n 1 & 0 \\\\\n 2 & 3\n\\end{pmatrix}"

What is B^-1?

 Show that the inverse of a square matrix A exists if and only if the

eigenvalues λ1

,λ2

,··· ,λn of A are different from zero. If A

−1

exists

show that its eigenvalues are 1

λ1

,

1

λ2

,···

1

λn

.

Let T : R3 → R3 be defined by T (x1

, x2

, x3

) = (x1

, x2

,−x1 − x2

). Find a

matrix which represents T

Suppose T€L(R^2) is defined by T(x, y) =(-3y,x).find the eigenvalues of T

Suppose T€ L(V ) is invertible.

(a) Suppose lemtha € F with lemtha not equal to 0. Prove that lemtha is an eigenvalue of T if and only if 1/lemtha is an eigenvalue of T^-1.

(b) Prove that T and T^-1 have the same eigenvectors.

Suppose V is the finite-dimensional and S; T€ L(V ). Prove that ST and T S have the same eigenvalues.

Check whether each of the following subsets of R 3 is linearly independent. i) {(1,2,3),(−1,1,2),(2,1,1)}. ii) {(3,1,2),(−1,−1,−3),(−4,−3,0)