Question #113459
Define the determinant function and write its properties. Also use this definition
Find the determinant of a 3 × 3 matrix.
1
Expert's answer
2020-05-06T16:02:42-0400

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\sum_{{1<j<n}} (-1)i+j a1,j det A1,j where A1,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:


123412213\begin{vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 2 & 1 & 3 \\ \end{vmatrix} = 11213\begin{vmatrix} 1 & 2 \\ 1 & 3 \end{vmatrix} -24223\begin{vmatrix} 4 & 2 \\ 2 & 3 \end{vmatrix} +34121\begin{vmatrix} 4 & 1 \\ 2 & 1 \end{vmatrix} = 1-2(8)+3(2) = 1-16+6 = -9



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