Calvin encoded a secret phrase using matrix multiplication. Using A = 1, B = 2, C = 3, and so on, he multiplied the clear text code for each letter by the matrix c=3 -2 4 -3 to get a matrix that represents the encoded text. The matrix representing the encoded text is...... What is the secret phrase? Determine the location of spaces after you decode the text.

Our matrix C is the encoder.  Let X be our secret message and let B be our encoded message.  Therefore the relationship between the 3 can be represented as:

CX = B

So we want to solve for X (our secret message). So using matrix algebra:

X = C^-1*B
where C^-1 is the inverse of C.

C^-1 = | 3 -2 |
           |4 -3 |

So we take C^-1 and multiply it by matrix B to get X.  Matrix B is already given to us.  So just multiply those two.  I used a calculator to get:

X = | 20 8 5 3 15 12 15 |
      | 18 9 19 2 12 21 5 |

And since A = 1, B = 2, C = 3, etc....

We decode our message and it says:

X = | T H E C O L O |
      | R I S B L U E |

So the answer is: The color is blue (D)