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

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