High School Mathematics Extensions/Matrices/Problem Set

High School Mathematics Extensions

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Supplementary Chapters — Primes and Modular Arithmetic — Logic

Mathematical Proofs — Set Theory and Infinite Processes — Counting and Generating Functions — Discrete Probability

Matrices — Further Modular Arithmetic — Mathematical Programming — Markov Chains

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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, A ≠ 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