# Matrices

## Determinant & Inverse of a 2x2 Matrix

The determinant of a 2x2 matrix:

The inverse:

The inverse of a matrix only exists where

## Minors & Cofactors

• There is a matrix minor corresponding to each element of a matrix
• The minor is calculated by
• ignoring the values on the current row and column
• calculate the determinant of the remaining 2x2 matrix

Example:

The minor of the top left corner is:

The cofactor is the minor multiplied by it's correct sign. The signs form a checkerboard pattern:

The matrix of cofactors is denoted .

## Determinant of a 3x3 Matrix

The determinant of a 3x3 matrix is calculated by multiplying each element in one row/column by it's cofactor, then summing them. For the matrix:

This shows the expansion of the top row, but any column or row will produce the same result.

## Inverse of a 3x3 Matrix

• Calculate matrix of minors
• Calculate matrix of cofactors
• Transpose
• Multiply by 1 over determinant

### Example

The transposed matrix of cofactors is therefore:

Explanding by the bottom row to calculate the determinant (it has 2 zeros so easy calculation):

Calculating inverse: