Linear Algebra Answers

(4.3) Let ~u =< 0, 1, 1 >, ~v =< 2, 2, 0 > and w~ =< −1, 1, 0 > be three vectors in standard form.

(a) Determine which two vectors form a right angle triangle?

(b) Find θ := ~ucw~ , the angel between the given two vectors. (2)

(4.4) Let x < 0. Find the vector ~n =< x, y, z > that is orthogonal to all three vectors (2) ~u =< 1, 1, −2 >, ~v =< −1, 2, 0 > and w~ =< −1, 0, 1 >.

(4.5) Find a unit vector that is orthogonal to both ~u =< 0, −1, −1 > and ~v =< 1, 0, −1 >.

Group or not group? The set of M_nxn (R) of all nxn matrices under multiplication.

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