Show that A and B are not similar matrices A=begin{bmatrix}1 & 0 &1 2 & 0 &2 3&0&3 end{bmatrix} , B=begin{bmatrix}1 & 1 &0 2 & 2 &0 0&1&1 end{bmatrix}

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Write the solution set of the given homogeneous system in parametric vector form.
${\displaystyle x_1+3x_2-5x_3=0}$
${\displaystyle x_1+4x_2-8x_3=0}$
${\displaystyle -3x_1-7x_2+9x_3=0}$

Find values for the variables so that the matrices are equal.
$\left[\begin{array}{c}x\\ 7\end{array}\right]=\left[\begin{array}{c}11\\ y\end{array}\right]$

Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and identify its principal and secondary diagonals.
$\left[\begin{array}{ccc}1& 0& -i\\ 0& -2& 4-i\\ i& 4+i& 3\end{array}\right]$
$\left[\begin{array}{ccc}7& 0& 4\\ 0& -2& 10\\ 4& 10& 5\end{array}\right]$

$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 4& 1\\ 0& 2& 1\end{array}\right]$
Find $C_{21}\text{ and }{C}_{13}$ , where C=2A-3B

In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.