Matrix multiplication is a row-by-column multiplication where each element of one matrix is multiplied by every element of another matrix. Matrices to be multiplied do not need to be of the same order, by definition the number of columns of the first matrix must equal the number of rows of the second matrix, otherwise all row elements of the first matrix could not be multiplied by a corresponding column element of the second matrix and the multiplication would be undefined.

A formal definition of matrix multiplication follows.

If A = [aij] is an m × n matrix and B = [bij] is an n × p matrix, the product AB is an m × p matrix.

AB = [Cij] where Cij = a1j b1j + ai2 b2j + ai3 b3j + . . . + ain bnj

If we were to multiply matrix A and matrix B:

Matrix A with Order 3 × 3

3×3 | Column A1 | Column A2 | Column A3 |

Row A1: | 1 | 0 | 4 |

Row A2: | 2 | 3 | 5 |

Row A3: | -3 | 4 | -2 |

×

Matrix B with Order 3 × 2

3×2 | Column B1 | Column B2 |

Row B1: | -1 | 4 |

Row B2: | 3 | 2 |

Row B3: | 5 | 0 |

The product of matrices A and B can be written as:

Row A1 × Column B1

1 (-1) + 0 (3) + 4 (5)

Row A2 × Column B1

2 (-1) + 3 (3) + 5 (5)

Row A3 × Column B1

-3 (-1) + 4 (3) + -2 (5)

Row A1 × Column B2

1 (4) + 0 (2) + 4 (0)

Row A2 × Column B2

2 (4) + 3 (2) + 5 (0)

Row A3 × Column B2

-3 (4) + 4 (2) + -2 (0)

The resulting matrix AB with Order 3 × 2 is:

Product Matrix AB

3×2 | Column 1 | Column 2 |

Row 1: | 19 | 4 |

Row 2: | 32 | 14 |

Row 3: | 5 | -4 |

Let A, B and C be matrices and let c be a scalar.

- A (BC) = (AB) C Associative Property of Multiplication
- A (B + C) = AB + AC Distributive Property
- (A + B) C = AC + BC Distributive Property
- c (AB) = (cA) B = A (cB) Associative Property Scalar Multiplication
- AB ≠ BA If AB = BA they are inverse matrices (Not singular matrices)

The identity matrix is any n × n square matrix that consists of 1’s on its main diagonal and 0’s elsewhere. It is denoted by:

In =

4×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | 1 | 0 | 0 | 0 |

Row 2: | 0 | 1 | 0 | 0 |

Row 3: | 0 | 0 | 1 | 0 |

Row 4: | 0 | 0 | 0 | 1 |

The identity matrix has the property that AIn = A and InA = A thus,

Matrix A

3×3 | Column 1 | Column 2 | Column 3 |

Row 1: | 2 | -5 | 4 |

Row 2: | 1 | 0 | -3 |

Row 3: | 6 | 4 | 2 |

×

Identity Matrix

3×3 | Column 1 | Column 2 | Column 3 |

Row 1: | 1 | 0 | 0 |

Row 2: | 0 | 1 | 0 |

Row 3: | 0 | 0 | 1 |

=

Resulting Matrix AIn

3×3 | Column 1 | Column 2 | Column 3 |

Row 1: | 2 | -5 | 4 |

Row 2: | 1 | 0 | -3 |

Row 3: | 6 | 4 | 2 |

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