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= (aij) where i and j lie between 1 and n.
∑1<j<n (-1)i+j a1,j det A1,j where A1,j is a sub-matrix of P.
properties of determinant:
- the determinant does not change if rows are changed into columns or columns are changed into rows.( reflection property)
- if all the elements of a row or a column are zero, then determinant of the matrix is zero (zero property)
- the interchanging of any two rows of a matrix changes the sign.( switching property)
- if the row or a column are multiplied by a scale constant then determinant gets multiplied by same scalar constant.( scalar multiple property)
- if all the elements of a row/column are same, then determinant is zero.(proportionality)
determinant of 3x3 matrix:
∣∣142211323∣∣ = 1∣∣1123∣∣ -2∣∣4223∣∣ +3∣∣4211∣∣ = 1-2(8)+3(2) = 1-16+6 = -9
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