Find the rank by theorem "Rank in Terms of Determinants " and check by row reduction. (Show details.)

$$\left[\begin{array}{rrr}2 & 1 & 0 \\ 13 & -13 & 12 \\ -3 & 5 & -4\end{array}\right]$$

Calculate the following expression (showing the details of your work) or indicate why they do not exist, when

$$\mathbf{A}=\left[\begin{array}{rrr} 9 & 2 & 8 \\ 2 & 18 & 10 \\ 8 & 10 & 15 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{rrr} 0 & 2 & 6 \\ -2 & 0 & -3 \\ -6 & 3 & 0 \end{array}\right]$$

$$\mathbf{a}=\left[\begin{array}{l} 3 \\ 7 \\ 1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right]$$

$\mathbf{A}-\mathbf{A}^{\mathbf{T}}$

Use the explicit formula for the reverese.

$$\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$$

(a) Show experimentall that the n x n matrix A=[ajk] with ajk=j+k-1 has rank 2 for any n.(n=4). Try to prove it. (b) Do the same when ajk=j+k+c, where c is any positive integer. (c) What is rank A if ajk=2^j+k-2? Try to find other large matrices of low rank independent of n.

If U1, U2 are upper triangular and L1, L2 are lower triangular, which of the following are triangular? U1+U2, U1U2, U2²1, U1+L1, U1L1, L1+L2

Are the following sets of vectors linearly independent? Show the details of your work. [2 0 0 7], [2 0 0 8], [2 0 0 9], [2 0 1 0]

Solve by Cramer's rule and check by Gauss elimination and back substitution. (Show details.)

$$\begin{aligned} & w+2 x \quad-3 z=30 \\ & 4 x-5 y+2 z=13 \\ & 2 w+8 x-4 y+z=42 \\ & 3 w \quad+y-5 z=35 \end{aligned}$$

Calculate the following expression (showing the details of your work) or indicate why they do not exist, when

$$\mathbf{A}=\left[\begin{array}{rrr} 9 & 2 & 8 \\ 2 & 18 & 10 \\ 8 & 10 & 15 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{rrr} 0 & 2 & 6 \\ -2 & 0 & -3 \\ -6 & 3 & 0 \end{array}\right]$$

$$\mathbf{a}=\left[\begin{array}{l} 3 \\ 7 \\ 1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right]$$

$\mathbf{A B}, \mathbf{B A}$

By definition, $A$ is idempotent if $\mathbf{A}^2=\mathbf{A}$, and $\mathbf{B}$ is nilpotent if $\mathbf{B}^m=\mathbf{0}$ for some positive integer $m$. Give examples (different from $\mathbf{0}$ or I). Also give examples such that $\mathrm{A}^2=\mathbf{I}$ (the unit matrix).

Find the inverse transformation. (Show the details of your work.)

$$\begin{aligned} & y_1=x_1+x_2-2 x_3 \\ & y_2=\quad x_1+x_2+2 x_3 \\ & y_3=-2 x_1+2 x_2+4 x_3 \end{aligned}$$

Is the given set (taken with the usual addition and scalar multiplication) a vector space? (Give a reason.) If your answer is yes. find the dimension and a basis.

All vectors in $R^3$ satisfying $2 v_1+3 v_2-v_3=0$, $v_1-4 v_2+v_3=0$

Find the rank and a basis for the row space and for the column space. Hint. Row-reduce the matrix and its transpose. (You may omit obvious factors from the vectors of these bases.)

$$\left[\begin{array}{rrr}8 & 2 & 5 \\ 16 & 6 & 29 \\ 4 & 0 & -7\end{array}\right]$$

Find the inverse by Gauss-Jordan or state that it does not exist.

Check by using

$$\mathbf{A} \mathbf{A}^{-1}=\mathbf{A}^{-1} \mathbf{A}=\mathbf{I}$$

$$\left[\begin{array}{rr}0.6 & 0.8 \\ 0.8 & -0.6\end{array}\right]$$

Solve the following system or indicate the nonexistence of solutions. (Show the details of your work.)

$$\begin{array}{r} 3.0 x+6.2 y=0.2 \\ 2.1 x+8.5 y=4.3 \\ \end{array}$$