Determinants

A determinant is a function of the components of a square matrix.

A determinant calculation results in a scalar.

Below, matrix A is a 2×2 matrix and the notation |A| signifies the determinant of A.

$A = \begin{bmatrix} a && b \\ c && d \end{bmatrix}$

$|A| = ad - bc$

Example:

$A = \begin{bmatrix} 1 && 2 \\ e && 4 \end{bmatrix} = (1)(4) - (2)(3) = -2$

Some Properties of Determinants

We won’t show any proofs for properties here. Rather, we will use the above matrix with a,b,c,d to illustrate a few properties.

$B = \begin{bmatrix} c && d \\ a && b \end{bmatrix} = cb - da$

$= - ( -(cb) + (da)$

$= - (da - cb)$

$= - (ad - bc)$

$= - |A|$

Interchanging the two rows multiplied the determinate by -1.

You can show that the same thing occurs if you interchange two columns.