# 0.3 Matrices: homework  (Page 2/4)

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$x+y+2z=0$
$x+2y+z=0$
$2x+3y+3z=0$

Find three solutions to the following system of equations.

$x+2y+z=\text{12}$
$y=3$

(5, 3, 1), (4, 3, 2) (3, 3, 3)

For what values of $k$ the following system of equations have a). No solution? b). Infinitely many solutions?

$x+2y=5$
$2x+4y=k$
$x+3y-z=5$

( $5-3s+t$ , $s$ , $t$ )

Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.

## Inverse matrices

In the next two problems, verify that the given matrices are inverses of each other.

$\left[\begin{array}{cc}7& 3\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& -3\\ -2& 7\end{array}\right]$
$\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 2& 3& -4\end{array}\right]\left[\begin{array}{ccc}3& -4& 1\\ 2& -4& 1\\ 3& -5& 1\end{array}\right]$

In the following problems, find the inverse of each matrix by the row-reduction method.

$\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$
$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$
$\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]$
$\left[\begin{array}{ccc}1& 1& -1\\ 1& 0& 1\\ 2& 1& 1\end{array}\right]$
$\left[\begin{array}{ccc}1& 2& –1\\ –1& –3& 2\\ –1& –1& 1\end{array}\right]$
$\left[\begin{array}{ccc}1& 1& 1\\ 3& 1& 0\\ 1& 1& 2\end{array}\right]$

In the following problems, first express the system as AX = B, and then solve using matrix inverses found in the preceding four problems.

$3x-5y=2$
$-x+2y=0$

(4, 2)

$\begin{array}{ccccccc}x& +& & & 2z& =& 8\\ & & y& +& 4z& =& 8\\ & & & & z& =& 3\end{array}$
$\begin{array}{ccccccc}x& +& y& -& z& =& 2\\ x& +& & & z& =& 7\\ 2x& +& y& +& z& =& \text{13}\end{array}$

(3, 3, 4)

$\begin{array}{ccccccc}x& +& y& +& z& =& 2\\ 3x& +& y& & & =& 7\\ x& +& y& +& 2z& =& 3\end{array}$

Why is it necessary that a matrix be a square matrix for its inverse to exist? Explain by relating the matrix to a system of equations.

If a matrix $M$ has an inverse, then the system of linear equations that has $M$ as its coefficient matrix has a unique solution. If a system of linear equations has a unique solution, then the number of equations must be the same as the number of variables. Therefore, the matrix that represents its coefficient matrix must be a square matrix.

Suppose we are solving a system $\text{AX}=B$ by the matrix inverse method, but discover $A$ has no inverse. How else can we solve this system? What can be said about the solutions of this system?

## Application of matrices in cryptography

In the following problems, the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27.

 A B C D E F G H I J K L M 1 2 3 4 5 6 7 8 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26

In the next two problems, use the matrix A, given below, to encode the given messages.

$A=\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$

In the two problems following, decode the messages that were encoded using matrix A.

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

Encode the message: WATCH OUT!

$\left[\begin{array}{c}\text{71}\\ \text{24}\end{array}\right]\left[\begin{array}{c}\text{66}\\ \text{23}\end{array}\right]\left[\begin{array}{c}\text{78}\\ \text{35}\end{array}\right]\left[\begin{array}{c}\text{87}\\ \text{36}\end{array}\right]\left[\begin{array}{c}\text{114}\\ \text{47}\end{array}\right]$

Encode the message: HELP IS ON THE WAY.

Decode the following message:

64 23 102 41 82 32 97 35 71 28 69 32

RETURN HOME

Decode the following message:

105 40 117 48 39 19 69 32 72 27 37 15 114 47

In the next two problems, use the matrix B, given below, to encode the given messages.

$B=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 2\\ 1& 0& -1\end{array}\right]$

In the two problems following, decode the messages that were encoded using matrix $B$ .

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space(s) if necessary.

Encode the message using matrix $B$ :

$\left[\begin{array}{c}\text{12}\\ \text{51}\\ 9\end{array}\right]\left[\begin{array}{c}\text{11}\\ \text{67}\\ 2\end{array}\right]\left[\begin{array}{c}\text{19}\\ \text{95}\\ \text{14}\end{array}\right]\left[\begin{array}{c}\text{14}\\ \text{105}\\ -\text{11}\end{array}\right]\left[\begin{array}{c}\text{15}\\ \text{87}\\ -3\end{array}\right]\left[\begin{array}{c}\text{27}\\ \text{91}\\ \text{18}\end{array}\right]\left[\begin{array}{c}4\\ \text{67}\\ -\text{23}\end{array}\right]$

Encode the message using matrix $B$ :

MAY THE FORCE BE WITH YOU.

Decode the following message that was encoded using matrix B:

8 23 7 4 47 –2 15 102 –12 20 58 15 27 80 18 12 74 –7

Decode the following message that was encoded using matrix B:

12 69 –3 11 53 9 5 46 –10 18 95 –9 25 107 4 27 76 22 1 72 –26

how do you translate this in Algebraic Expressions
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