# PROPERTIES OF DETERMINANTS

Property 1: Let A = [aij] be a square matrix of order  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, $$\sum\limits_{j=1}^{n}{{{a}_{ij}}{{C}_{ij}}}=\left| A \right|$$ and $$\sum\limits_{i=1}^{n}{{{a}_{ij}}{{C}_{ij}}}=\left| A \right|$$

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.

$$\sum\limits_{j=1}^{n}{{{a}_{ij}}{{C}_{kj}}=0}$$ and $$\sum\limits_{i=1}^{n}{{{a}_{ij}}{{C}_{ik}}=0}$$

Property 3: Let A = [aij] be a square matrix of order n then |A|=|AT|.

Ex: $$\left|\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left|\begin{matrix}{{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\{{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\{{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right|$$

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|.

Ex: $$\left|\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}}\\\end{matrix}\right|=-\left|\begin{matrix}{{a}_{1}} & {{c}_{1}} & {{b}_{1}} \\{{a}_{2}} & {{c}_{2}} & {{b}_{2}} \\{{a}_{3}} & {{c}_{3}} & {{b}_{3}}\\\end{matrix}\right|$$

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.

Ex: $$\left|\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix}\right|=0$$ 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|

Ex: $$\left| \begin{matrix}{{a}_{1}} & k{{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & k{{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & k{{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left| \begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|$$ 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|

Property 7: If each element of a row (column) of a determinant is a sum of two terms, then determinant can be written as sum of two determinants.

Ex: $$\left| \begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}}+{{d}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}}+{{d}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}}+{{d}_{3}} \\\end{matrix} \right|=\left| \begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|+\left| \begin{matrix}{{a}_{1}} & {{b}_{1}} & {{d}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{d}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{d}_{3}} \\\end{matrix} \right|$$

Property 8: If each element of a row (column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column), then the value of the determinant remains same.

Ex: $$\left| \begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|=\left| \begin{matrix}{{a}_{1}} & {{b}_{1}}+2{{a}_{1}}+3{{c}_{1}} & {{c}_{1}} \\{{a}_{2}} & {{b}_{2}}+2{{a}_{2}}+3{{c}_{2}} & {{c}_{2}} \\{{a}_{3}} & {{b}_{3}}+2{{b}_{3}}+3{{c}_{3}} & {{c}_{3}} \\\end{matrix} \right|$$

Obtained after C₂ → C₂ + 2C₁ + 3C₃

Property 9: If each element of a row (column) of a determinant zero, then its value is zero.

Ex: $$\left| \begin{matrix}0 & 0 & 0 \\{{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\{{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right|=0$$ and $$\left| \begin{matrix}{{a}_{1}} & 0 & {{c}_{1}} \\{{a}_{2}} & 0 & {{c}_{2}} \\{{a}_{3}} & 0 & {{c}_{3}} \\\end{matrix} \right|=0$$

Property 10: If A = [aij] is a diagonal matrix of order n (≥ 2), then |A|= a₁₁, a₂₂, a₃₃ … anm.

Property 11: If A and B are square matrices of the same order, then |AB|=|A||B|.

Property 12: If A = [aij] is triangular matrix of order n, then |A|= a₁₁, a₂₂, a₃₃ … anm.

Ex: $$\left| \begin{matrix}{{a}_{1}} & 0 & 0 \\{{a}_{2}} & {{b}_{2}} & 0 \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix} \right|={{a}_{1}}{{b}_{2}}{{c}_{3}}$$ and $$\left| \begin{matrix}{{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\0 & {{b}_{2}} & {{b}_{3}} \\0 & 0 & {{c}_{3}} \\\end{matrix} \right|={{a}_{1}}{{b}_{2}}{{c}_{3}}$$

Property 13: If A is a non-singular square matrix of order n, then |adj A| = |A|n-1.

Property 14: If A is a skew-symmetric matrix of odd order, then |A| = 0.

Property 15: If A is a skew-symmetric matrix of even order, then |A| is a perfect square.

$$\left| \begin{matrix}1 & 1 & 1 \\a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\\end{matrix} \right|$$ = (a – b) (b – c) (c – a)

$$\left| \begin{matrix}1 & 1 & 1 \\a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\\end{matrix} \right|$$ = (a – b) (b – c) (c – a) (a + b + c)