# Introduction

In mathematics, a matrix is a group of numbers, symbols, or expressions (often referred to as entries) arranged in a rectangular fashion. The group of numbers is shown as a matrix by being placed in rows and columns between two braces. Every matrix has a dimension, such as, 2x3, the first number represents the number of rows and the second represents the number of columns. There are no limitations upon the dimensions of the matrix, however, they do place limitations upon operations.

J = | ||||
---|---|---|---|---|

2 | -5 | 14 | 23 | -9 |

-4 | 19 | 4 | 34 | 9 |

41 | 5 | 304 | 1 | -6 |

The matrix as a whole is given a symbol to represent it, in this case it is J, this allows you to name the values within the matrix fairly easily. For example, the 2 within the matrix would be j1,1 because it is in the first row and first column. Another example would be j2,3, would would represent the number 4 within the set.

# Construction

In precalculus, we use matrices as a way to solve a system of linear equations, such as:

Examples of Systems of Linear Equations | |
---|---|

8a - b = 9; 4a + 9b = 7 | 3a - b + 14c = 7; 2a + 2b + 3c = 0; a - 12b - 18c = 33 |

3r + s - 7t = 15; r - 12s + t = 0; 5s - 4t = 8 | x + 7y = 9; 2x - y = 18 |

For the now on let’s focus on the second set of equations, in order to arrange these values within the matrix we dedicate each row to a specific function and each column to the coefficients of each variable.

# Multiplication

Matrices can be multiplied together, however, there is a certain restriction, that being that the number of columns in matrix A must be equal to the number of rows in matrix B - in other words, the inside dimensions of the two matrices must be the same. When multiplying, you have to multiply the first row in matrix A with the first column in matrix B, the second row in matrix A with the second column in matrix B, and so on - it cannot be done in a different order.

Example:

X = | Y = | ||||
---|---|---|---|---|---|

1 | 2 | 1 | 2 | 3 | |

3 | 4 | 4 | 5 | 6 | |

5 | 6 |

Z = | ||
---|---|---|

1×1 + 2×4 | 1×2 + 2×5 | 1×3 + 2×6 |

3×1 + 4×4 | 3×2 + 4×5 | 3×3 + 4×6 |

5×1 + 6×4 | 5×2 + 6×5 | 5×3 + 6×6 |

Z = | ||
---|---|---|

9 | 12 | 15 |

19 | 26 | 33 |

29 | 40 | 51 |

# Solving Systems Using Inverse Matrix

Inverse matrices can be used to solve systems of equations by the expression AA^{-1} = A^{-1}A = I, where the number of variables within the equation are equal to the number of variables. For example, if matrix A is (4, 3, 3, 2), then since those two matrices will have to combine to equal the identity matrix, which in this case (for 2x2 square matrices) is: (1, 0, 0, 1), then matrix A^{-1} must be (2, 3, 3, -4), since that is the inverse of matrix A and more importantly what is necessary to make it into the identity matrix.

# Sum and Difference

When finding the sum of two matrices, both matrices must have the exact same dimensions - meaning that a 2x3 matrix must be added to another 2x3 matrix. To solve, add the term in the first column of the first row (the order does matter) of matrix A with the corresponding term in matrix B. Then, continue adding corresponding terms until the matrix is solved. For subtracting matrices, the same process is used, where the only difference is that the term in matrix B, of course, is subtracted from the corresponding value in matrix A.

Let’s look at an example using matrix addition:

A = | B = | |||
---|---|---|---|---|

-1 | 3 | 0 | 6 | |

5 | 7 | 2 | -5 |

C = | |
---|---|

-1 + 0 | 3 + 6 |

5 + 2 | 7 - 5 |

C = | |
---|---|

-1 | 9 |

7 | 2 |