Matrix Determinant

Finding determinants of 3x3 and more matrices

Cofactor Method

+-+-+-+-.....

-+-+-+-+.....

.....
 * 1) Choose a row/column
 * 2) Cross it out corresponding column(if row picked) or row(if column picked) to create smaller matrix(2x2 if 3x3)
 * 3) Find determinant of smaller matrices
 * 4) Multiply determinants with +-+/-+- signs as shown above and also multiply is by the elements of row/column cross out(if row/column crossed out was 1,2,3,4 multiply each of the determinants with 1,2,3,4)
 * 5) Sum the above.

Practical Row Operation method

Get the row echelon form. Achieve upper triangular form, meaning anything below the main diagonal is a zero. The diagonal doesn't have to be only composed of ones.

Factors affecting determinant value

Multiplying a row by a non zero number multiplies the determinant by the same number. Multiply row by k -> multiply determinant by k.

Switching the rows around multiplies the determinant by -1.
 * 1) Reduce the matrix to row echelon form. Just form a diagonal.
 * 2) Make appropriate changes to matrix (If row is multiplied by 5, divide entire matrix by 5. If row is switched around, multiply entire matrix by -1)to make up for shifting rows around and multiplying rows by non-zero values.
 * 3) Multiply the value of the diagonal and the values outside the matrix to find the determinant

if determinant == zero ~> Matrix =/ invertible

if matrix =/ invertible ~> unique non-trivial solution

if determinant =/ zero ~> Matrix == invertible

If matrix == invertible ~> trivial solutions and linearly independent