Lecture on Properties of Determinants
Content covered: 
  • Important properties of determinant are:
  1. The value of Δ does not change if rows and columns are interchanged.
    |AT|=|A|
    \( \begin{vmatrix}2&3\\4&5\end{vmatrix} \) = \( \begin{vmatrix}2&4\\3 &5\end{vmatrix} \)
  2. If any row or column of a matrix is 0 then |A|=0
    Δ of
    \( \begin{vmatrix}0&0&0&0\\7&5&9&6\\5&3&2&1\\7&6&2&5\end{vmatrix} \) =0
  3. If matrix have identical rows or column . The value of Δ =0. 
    Δ of
    \( \begin{vmatrix}6&5&3\\2&9&7\\6&5&3\end{vmatrix} \) =0
  4. The value of a Δ is unchanged if row or column are added m times.
    A= \( \begin{vmatrix}7&6\\2 &1\end{vmatrix} =-5 \) 
    After R1<-R1+2R2
    \( \begin{vmatrix}11&8\\2 &7\end{vmatrix} =-5 \)
  5. Multiplication with scalar
    A=\( \begin{vmatrix}a1&a2\\ a3 &a4\end{vmatrix}
    \)
     
    A'= \( \begin{vmatrix}k a1&k a2\\ a3 &a4\end{vmatrix} =k|A|\)
  6. Product with cofactor
    \( \begin{vmatrix}a1&a2\\a3 &a4\end{vmatrix} =k^n|A| \) where n is order of Δ
    a1*cf(a1)+a2*cf(a2)=|A|
    a1*cf(a3)+a2*cf(a4)=0
  7. Determinant of multiplication of two matrices =Determinant of first matrix X Determinant of second matrix
      |AB|=|A| X |B|
  8. |Adj A|=|A|n-1
    Adj A=[cofactor A]T
    A .adjA=|A|.I
    |Adj A|=|A|.I/A=|A|n-1 
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2Comments
Sudipta Halder @sudiptahalder 24 Nov 2014 02:00 am
JYOTI @jyotinagwan 3 Mar 2015 04:00 pm