# High School Mathematics Extensions/Matrices/Problem Set

Content HSME Matrices Recurrence Relations Problem Set Project Exercises Solutions Problem Set Solutions Definition Sheet Full Version

## Problem Set

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

$X = \begin{pmatrix} ?&?&?&?\\ ?&?&?&?\end{pmatrix}$,

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

$\begin{pmatrix} 2&3\\ 3&5\end{pmatrix} \begin{pmatrix} ?&?&?&?\\ ?&?&?&?\end{pmatrix} =\begin{pmatrix} 28&94&70&102\\ 44&153&112&163\end{pmatrix}$.

What was Zhuo's message to Jenny?

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

$A\begin{pmatrix}1 \\ 2 \end{pmatrix}= \begin{pmatrix}1 \\ 0 \end{pmatrix}$

and $A\begin{pmatrix}3 \\ 4 \end{pmatrix}= \begin{pmatrix}0 \\ 1 \end{pmatrix}$.

What is the inverse of A?

3. Let

$J = \begin{pmatrix}0 &1\\ 1 &0 \end{pmatrix}$,

and let K = I + J. Show that Kn = nK.

4. Suppose

$A^k = 0$

and

$B^l = 0$

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

$(A + B)^m = 0$

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

$A\begin{pmatrix} p\\ q \end{pmatrix}= \begin{pmatrix} p\\ q \end{pmatrix}$

6. Find A such that: $A^3 = \begin{pmatrix} -10&18\\ -9&17 \end{pmatrix}$

...more to come. Please contribute good problems