Determinant Matrix Problems – I
1. Show that \(\left| \begin{matrix}a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\{{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\\end{matrix} \right|\) = abc (a – b) (b – c) (c – a)
Solution: Given matrix
\(\left| \begin{matrix}a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\{{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\\end{matrix} \right|\)
Take a common abc three rows
\(=abc\left| \begin{matrix}1 & 1 & 1 \\a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\\end{matrix} \right|\)
We can apply elementary column transformations
C₁ → C₁ – C₂
C₂ → C₂ – C₃
\(=abc\left| \begin{matrix}0 & 0 & 0 \\a-b & b-c & c \\{{a}^{2}}-{{b}^{2}} & {{b}^{2}}-{{c}^{2}} & {{c}^{2}} \\\end{matrix} \right|\)
Take a common (a – b) (b – c)
\(=abc(a-b)(b-c)\left| \begin{matrix}0 & 0 & 0 \\1 & 1 & 1 \\a+b & b+c & {{c}^{2}} \\\end{matrix} \right|\)
= abc (a – b) (b – c) [b + c – a – b]
= abc (a – b) (b – c) (c – a)
Hence proved \(\left| \begin{matrix}a & b & c \\{{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\{{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\\end{matrix} \right|\) = abc (a – b) (b – c) (c – a)
2. Without expanding the determinant, show that \(\left| \begin{matrix}a & {{a}^{2}} & bc \\b & {{b}^{2}} & ca \\c & {{c}^{2}} & ab \\\end{matrix} \right|=\left| \begin{matrix}1 & {{a}^{2}} & {{a}^{3}} \\1 & {{b}^{2}} & {{b}^{3}} \\1 & {{c}^{2}} & {{c}^{3}} \\\end{matrix} \right|\)
Solution: Given matrix
\(\left| \begin{matrix}a & {{a}^{2}} & bc \\b & {{b}^{2}} & ca \\c & {{c}^{2}} & ab \\\end{matrix} \right|\)
Take a common abc 3rd column
\(\left| \begin{matrix}a & {{a}^{2}} & bc \\b & {{b}^{2}} & ca \\c & {{c}^{2}} & ab \\\end{matrix} \right|=abc\left| \begin{matrix}a & {{a}^{2}} & \frac{1}{a} \\b & {{b}^{2}} & \frac{1}{b} \\c & {{c}^{2}} & \frac{1}{c} \\\end{matrix} \right|\)
\(=\left| \begin{matrix}{{a}^{2}} & {{a}^{3}} & 1 \\{{b}^{2}} & {{b}^{3}} & 1 \\{{c}^{2}} & {{c}^{3}} & 1 \\\end{matrix} \right|\)
\(=\left| \begin{matrix}1 & {{a}^{2}} & {{a}^{3}} \\1 & {{b}^{2}} & {{b}^{3}} \\1 & {{c}^{2}} & {{c}^{3}} \\\end{matrix} \right|\)
Hence proved \(\left| \begin{matrix}a & {{a}^{2}} & bc \\b & {{b}^{2}} & ca \\c & {{c}^{2}} & ab \\\end{matrix} \right|=\left| \begin{matrix}1 & {{a}^{2}} & {{a}^{3}} \\1 & {{b}^{2}} & {{b}^{3}} \\1 & {{c}^{2}} & {{c}^{3}} \\\end{matrix} \right|\)