Matrix Operations

Study Notes

Matrix Operations

Matrix Addition

Two matrices can be added if they have the same dimension (number of rows and columns)
Addition is done element-wise, where corresponding elements are added together
Example:

A = | 1 2 |   B = | 3 4 |
    | 5 6 |       | 7 8 |

A + B =

    | 4 6 | 
    | 12 14 |

Matrix Subtraction

Similar to matrix addition, but corresponding elements are subtracted
Example:

A = | 1 2 |   B = | 3 4 |
    | 5 6 |       | 7 8 |

A - B =

    | -2 -2 | 
    | -2 -2 |

Matrix Multiplication

Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix
The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix
Example:

A = | 1 2 |   B = | 3 4 |
    | 5 6 |       | 5 6 |

A × B =

    | 19 22 |
    | 49 54 |

Matrix Scalar Multiplication

A matrix can be multiplied by a scalar (number)
Each element of the matrix is multiplied by the scalar
Example:

A = | 1 2 |   k = 3
    | 5 6 |

k × A =

    | 3 6 |
    | 15 18 |

Matrix Transpose

The transpose of a matrix is obtained by swapping its rows and columns
Denoted by A^T
Example:

A = | 1 2 |   A^T = | 1 5 |
    | 5 6 |       | 2 6 |

Matrix Inverse

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix
Denoted by A^(-1)
Example:

A = | 1 2 |   A^(-1) = | -2 1 |
    | 5 6 |       | 1.5 -0.5 |

Note: Not all matrices have an inverse. The inverse only exists if the matrix is square (has the same number of rows and columns) and has no zero rows or columns.

Description

This quiz covers the basics of matrix operations, including matrix addition, subtraction, multiplication, scalar multiplication, transpose, and inverse. Learn how to perform these operations and understand the rules and concepts behind them.