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Find three solutions to the following system of equations.
(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?
Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.
In the next two problems, verify that the given matrices are inverses of each other.
In the following problems, find the inverse of each matrix by the row-reduction method.
In the following problems, first express the system as AX = B, and then solve using matrix inverses found in the preceding four problems.
(3, 3, 4)
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?
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!
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 $B$ :
LUCK IS ON YOUR SIDE.
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
HEAD FOR THE HILLS
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
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