Matrices

Determinant & Inverse of a 2x2 Matrix

The determinant of a 2x2 matrix:

$a c b d = a d - b c$

The inverse:

$(a c b d)^{- 1} = \frac{1}{a d - b c} (d - c - b a)$

The inverse of a matrix $M$ only exists where $det M \neq = 0$

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:

$M = 320001 2 - 2 1$

The minor of the top left corner is:

$01 - 2 1 = 2$

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

$+ - + - + - + - +$

The matrix of cofactors is denoted $C$ .

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:

$M = a d h b e h c f i$

$det M = a \cdot e h f i - b \cdot d g f i + c \cdot d g e h$

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

$M^{- 1} = \frac{1}{det M} C^{T}$

$M = 320001 2 - 2 1$

$M_{11} = + 01 - 2 1 = 2 M_{12} = - 20 - 2 1 = - 2 M_{13} = + 2001 = 2$

$M_{21} = - 0121 = 2 M_{22} = + 3021 = 3 M_{23} = - 3001 = - 3$

$M_{31} = + 00 2 - 2 = 0 M_{32} = - 32 2 - 2 = 10 M_{33} = + 3200 = 0$

The transposed matrix of cofactors $C^{T}$ is therefore:

$C^{T} = 2 - 2 2 23 - 3 0100$

Explanding by the bottom row to calculate the determinant (it has 2 zeros so easy calculation): $det M = 0 \times 0 + 1 \times 10 + 0 \times 0 = 10$

Calculating inverse:

$M^{- 1} = \frac{1}{det M} C^{T} = \frac{1}{10} 2 - 2 2 23 - 3 0100 = 0.2 - 0.2 0.2 0.2 0.3 - 0.3 010$