Multiple part question: In multiplying two real numbers, we are familiar with the so-called "Zero-Product Property". Recall that this says that if you have two numbers x and y such that the product xy=0, then either x=0, y=0 or they are both zero. Can the same be said of matrices? In other words, given matrices A and B where AB=0, does that mean that either A=0, or B=0 or both (where 0 detonates the zero matrix here).

(a) Let A=|3 -6|.
|-1 2|
Try to construct a 2 x 2 matrix B such that AB is the zero matrix (make sure you clearly show the multiplication). Use two different nonzero columns for B.

(b) Suppose A=|1/2 0| and B=|-1 0|.
|0 1/2|. |0 -1|

(1) Compute the product AB.

(2) Compute the product BA.

(3) How do your answers in parts (1) and (2) compare? Will this be true in general for any two matrices when you multiply them (assuming their dimensions line up so that they may be multiplied)? If so, defend your reasoning. If not, then supply an example of two matrices C and D such that CD≠DC.

For this problem, you will be applying your knowledge of matrix multiplication and inverses. A cryptogram is a message written according to a secret code. For this particular cryptogram, we assign numbers to letters and characters as follows.

0=__ 12=L 24=X
1=A 13=M 25=Y
2=B 14=N 26=Z
3=C 15=O 27=.
4=D 16=P 28=,
5=E 17=Q 29=;
6=F 18=R 30=!
7=G 19=S 31=:
8=H 20=T 32='
9=I 21=U 33="
10=J 22=V 34=?
11=K 23=W 35=/

Suppose that you received the following numerical information via a top secret transmission.

(a) [67 118 -47]

(b) [41 79 -57]

(c) [81 159 -117]

(d) [59 97 -15]

(e) [47 85 -39]

(f) [58 109 -65]

(g) [131 261 -208]

Suppose also that you were sent the coding matrix A=[1 1 2 ]
[2 4 -3]
[3 6 -5 ]
The coding matrix can be thought of as the "key" to unlock the message. Use the coding matrix along with the given numerical information (a-g), and follow the process to decode the message. Be sure to include the decoded message. Include ALL work and show the steps taken to decode the message.

​