Answer to Question #113459 in Linear Algebra for elhussein

Question

Define the determinant function and write its properties. Also use this definition
Find the determinant of a 3 × 3 matrix.

Answer 1

The determinant of a function is a function from nxn square matrix to a scalar such that it is defined as:

let a matrix P be defined as P= (a_ij) where i and j lie between 1 and n.

$\sum_{{1<j<n}}$ (-1)^i+j a_1,j det A_1,j where A_1,j is a sub-matrix of P.

properties of determinant:

1. the determinant does not change if rows are changed into columns or columns are changed into rows.( reflection property)
2. if all the elements of a row or a column are zero, then determinant of the matrix is zero (zero property)
3. the interchanging of any two rows of a matrix changes the sign.( switching property)
4. if the row or a column are multiplied by a scale constant then determinant gets multiplied by same scalar constant.( scalar multiple property)
5. if all the elements of a row/column are same, then determinant is zero.(proportionality)

determinant of 3x3 matrix:

$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 2 & 1 & 3 \\ \end{vmatrix}$ = 1$\begin{vmatrix} 1 & 2 \\ 1 & 3 \end{vmatrix}$ -2$\begin{vmatrix} 4 & 2 \\ 2 & 3 \end{vmatrix}$ +3$\begin{vmatrix} 4 & 1 \\ 2 & 1 \end{vmatrix}$ = 1-2(8)+3(2) = 1-16+6 = -9