Matrix Operations

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.