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,

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

$X=\left(\begin{array}{cccc}?& ?& ?& ?\\ ?& ?& ?& ?\end{array}\right)$,

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

$\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)\left(\begin{array}{cccc}?& ?& ?& ?\\ ?& ?& ?& ?\end{array}\right)=\left(\begin{array}{cccc}28& 94& 70& 102\\ 44& 153& 112& 163\end{array}\right)$.

What was Zhuo's message to Jenny?

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

$A\left(\begin{array}{c}1\\ 2\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)$

and $A\left(\begin{array}{c}3\\ 4\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$.

What is the inverse of A?

3. Let

$J=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$,

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, A ≠ I (i.e. not equal to the identity) such that

$A\left(\begin{array}{c}p\\ q\end{array}\right)=\left(\begin{array}{c}p\\ q\end{array}\right)$

6. Find A such that: $A^3=\left(\begin{array}{cc}-10& 18\\ -9& 17\end{array}\right)$

...more to come. Please contribute good problems