High School Mathematics Extensions/Matrices/Problem Set

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,
 * 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 &times; 2 matrix, A &ne; I (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''