Properties of Determinants – 1
Property 1: Let A = [aij] be a square matrix of order n then the sum of the product of elements of any row (columns) with their cofactors is always equal to |A| or det (A) i.e, \(\sum\limits_{i=1}^{n}{{{a}_{ij}}{{C}_{ij}}}=\left| A \right|\) and \(\sum\limits_{j=1}^{n}{{{a}_{ij}}{{C}_{ij}}}=\left| A \right|\,\).
Property 2: Let A = [aij] be a square matrix of order n, then the sum of the product of elements of any row (column) with the cofactors of the corresponding elements of some other row (column) is zero.
\(\sum\limits_{i=1}^{n}{{{a}_{ij}}{{C}_{ik}}=0}\) and \(\sum\limits_{j=1}^{n}{{{a}_{ij}}{{C}_{kj}}=0}\).
Property 3: Let A = [aij] be a square matrix of order n then |A| = |AT|.
Example: \(\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right|\).
The value of a determinant remains unchanged if its rows and columns are interchanged.
Property 4: Let A = [aij] be a square matrix of order n (≥ 2) and let B be a matrix obtained from A by interchanging any two rows (columns) of A, then |B| = – |A|.
Example: \(\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=-\left| \begin{matrix} {{a}_{1}} & {{c}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{c}_{2}} & {{b}_{2}} \\ {{a}_{3}} & {{c}_{3}} & {{b}_{3}} \\\end{matrix} \right|\).
Property 5: If any two rows (columns) of a square matrix A = [aij] of order n (≥ 2) are identical. Then its determinant is zero i.e., |A| = 0.
Example: \(\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=0\) and \(\left| \begin{matrix} {{a}_{1}} & {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{a}_{2}} & {{b}_{2}} \\ {{a}_{3}} & {{a}_{3}} & {{b}_{3}} \\\end{matrix} \right|=0\)
Property 6: Let A = [aij] be a square matrix of order n, and let B be the matrix obtained from A by multiplying each element of a row (column) of A by scalar K, then |B| = k |A|.
Example: \(\left| \begin{matrix} {{a}_{1}} & k{{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & k{{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & k{{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|\),
And \(k\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left| \begin{matrix} k{{a}_{1}} & k{{b}_{1}} & k{{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|\),
Let A = [aij] be a square matrix of order n, then |kA| = kⁿ |A|.