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Matrix Algebra

By DarthVader

Date: 2023-04-24

Topic: 197 see comments

Post views: 845


Matrices

A matrix is a set of numbers enclosed in brackets.

Matrix Multiplication

You can only multiply two matrices if the second matrix has the same number of rows as the first matrix has columns.


Minors & Cofactors

A minor of an element in a matrix is the value of the determinant calculated from the rows and columns that do not contain that element.

The minor of an element together with its place sign is called the cofactor of that element.


Cofactor of a determinant:

The minor of an element together with its place sign is known as the cofactor of the element. For example, take the determinant below, and let it be known as | A | or "det A":

| 2 3 5 |

| 4 1 6 |

| 1 4 0 |

The minor of element 2 is given by:

| 1 6 |

| 4 0 |

and its value is 0 − 24 = −24 (calculated by: ad − bc, or; [ (1)(0) − (6)(4) ])

The place sign is positive for this element as seen here:

| + − + |

| − + − |

| + − + |

and so the cofactor is: +(−24) = −24.


Minor of a matrix element:

A minor of an element in a matrix is shown below for the element a11:

| a11 a12 a13 |

| a21 a22 a23 |

| a31 a32 a33 |

so the minor of the element a11 is:

| a22 a23 |

| a32 a33 |

The minor of the element a12 is shown below:

| a11 a12 a13 |

| a21 a22 a23 |

| a31 a32 a33 |

so the minor of the element a12 is:

| a21 a23 |

| a31 a33 |


Matrix Equations

For a pair of simultaneous equations E.G: ⇣

3x + 2y = 7

−x + y = 3

We can write this in matrix form as follows: ⇣

| 3 2 | | x | = | 7 |

|−1 1| | y | = | 3 |

A ” X ” “ B ”

 “ AX = B

Where the first matrix is A, the second matrix (the variable matrix) is X, and the third matrix is B..

So to solve this pair of simultaneous equations in matrix form, we must multiply both sides of the equation by the inverse matrix of matrix A … : ⇣

A-1AX = A-1B

This gives the answer: ⇣

[ X = A-1B ]

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Tags:

Matrix Algebra, Matrices, T194

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