# High School Mathematics Extensions/Matrices/Problem Set

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## Problem Set[edit]

1. Zhuo decided to write a message to Jenny using matrix encryption. He substituted each letter in the English alphabet with a number:

- A to 0
- B to 1
- C to 2
- ...
- Z to 25,

he then wrote his message in a 2 by 4 matrix as follows

- ,

now he pre-multiplies his secret message * X* with a matrix to get the result

- .

What was Zhuo's message to Jenny?

2. A 2 by 2 matrix * A* has the following property

and .

What is the inverse of * A*?

3. Let

- ,

and let K = I + J. Show that K^{n} = nK.

4. Suppose

and

prove or disprove that you can always find a positive integer *m* such that

5. Let *p* and *q* be two real numbers such that *p* + *q* = 1. Show that there is a 2 × 2 matrix, *A* ≠ *I* (i.e. not equal to the identity) such that

6. Find A such that:

*...more to come. Please contribute good problems*