# a) List all possible Jordan forms for 3 times 3 matrices. c) List all possible Jordan forms for 4 times 4 matrices.

Question
Matrices
a) List all possible Jordan forms for $$3 \times 3$$ matrices.
c) List all possible Jordan forms for $$4 \times 4$$ matrices.

2021-03-08
Step 1
Possible jordan forms of $$3 \times 3$$ matrices are:
$$\begin{bmatrix}* & 1 &0 \\ 0 & * &1 \\ 0&0&* \end{bmatrix} , \begin{bmatrix}* & 0 &0 \\ 0 & * &0 \\ 0&0&* \end{bmatrix}, \begin{bmatrix}* & 1 &0 \\ 0 & * &0 \\ 0&0&* \end{bmatrix}$$
Step 2
Possible jordan forms of $$4 \times 4$$ matrices are:
$$\begin{bmatrix}* & 0 &0&0 \\ 0 & * &0&0 \\ 0&0&*&0 \\0&0&0&* \end{bmatrix} ,\begin{bmatrix}* & 1 &0&0 \\ 0 & * &0&0 \\ 0&0&*&0 \\0&0&0&* \end{bmatrix} , \begin{bmatrix}* & 1 &0&0 \\ 0 & * &1&0 \\ 0&0&*&0 \\0&0&0&* \end{bmatrix} ,\begin{bmatrix}* & 1 &0&0 \\ 0 & * &1&0 \\ 0&0&*&1 \\0&0&0&* \end{bmatrix}$$

### Relevant Questions

Consider the "clock arithmetic" group $$(Z_{15}, \oplus)$$ a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group. b) List all of the subgroups of $$(Z_{15}, \oplus)$$

We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication.
$$\begin{bmatrix}1 & 4&3 \\0 & 1&4\\0&0&2 \end{bmatrix}\begin{bmatrix}3 & 2 \\1 & 1\\4&5 \end{bmatrix}$$
compute the indicated matrices (if possible). D+BC
Let $$A=\begin{bmatrix}3 & 0 \\ -1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 1 \\ 0 & 2 &3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 &6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\ -2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} , F=\begin{bmatrix}-1 \\ 2 \end{bmatrix}$$
compute the indicated matrices (if possible). B - C
Let
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\3 & 4\\5&6 \end{bmatrix}, D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix},E=\begin{bmatrix}4 & 2 \end{bmatrix},F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
The 2 \times 2 matrices A and B below are related to matrix C by the equation: C=3A-2B. Which of the following is matrix C.
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$

For the V vector space contains all $$\displaystyle{2}\times{2}$$ matrices. Determine whether the $$\displaystyle{T}:{V}\rightarrow{R}^{{1}}$$ is the linear transformation over the $$\displaystyle{T}{\left({A}\right)}={a}\ +\ {2}{b}\ -\ {c}\ +\ {d},$$ where $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$

Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{\#}=M^{-1}$$

Write the uncoded row matrices for the message.
Message: SELL CONSOLIDATED
Row Matrix Size: $$1 \times 3$$
Encoding Matrix: $$A=\begin{bmatrix}1 & -1&0 \\1 & 0&-1\\-6&3&2 \end{bmatrix}$$
SEL=$$\begin{bmatrix}19&5&12 \end{bmatrix}$$
L-C=$$\begin{bmatrix}12&0&3 \end{bmatrix}$$
ONS=$$\begin{bmatrix}15&14&19 \end{bmatrix}$$
OLI=$$\begin{bmatrix}15&12&9 \end{bmatrix}$$
DAT=$$\begin{bmatrix}4&1&20 \end{bmatrix}$$
ED=$$\begin{bmatrix}5&4&0 \end{bmatrix}$$
$$(i)\begin{bmatrix}1 & p & -1 \\ 2 & 1 & 7 \\ -3 & 3 & 2 \end{bmatrix} (ii) \begin{bmatrix}p & -1 & 7&2 \\ 2 & 1 & -5 & 3 \\ 1 & 3 & 2 & 0 \end{bmatrix}$$