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


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


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: