Question

show that determinants
1. a. bc. 1. a. a²
1. b. ca. = 1. b. b²
1. c. ab 1. c. c²



​

Answer 1

Appropriate Question :-

Prove that,

$\begin{gathered}\sf \left | \begin{array}{ccc}1&a&bc\\1&b&ca\\1&c& ab\end{array}\right | \end{gathered} = \begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\1&b& {b}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered}$

$\large\underline{\sf{Solution-}}$

Consider,

$\begin{gathered}\sf \left | \begin{array}{ccc}1&a&bc\\1&b&ca\\1&c& ab\end{array}\right | \end{gathered}$

$\boxed{\sf{  \:OP \: R_1 \to \: aR_1 \: }} \\ \\ \boxed{\sf{  \:\:OP \: R_2 \to \: bR_2 \: }} \\ \\ \boxed{\sf{  \:\:OP \: R_3 \to \: cR_3 \: }}$

So, using these operations, we get

$\rm \: = \: \dfrac{1}{abc} \begin{gathered}\sf \left | \begin{array}{ccc}a& {a}^{2} &abc\\b& {b}^{2} &bca\\c& {c}^{2} & cab\end{array}\right | \end{gathered}$

On taking abc from column 3, we get

$\rm \: = \: \dfrac{abc}{abc} \begin{gathered}\sf \left | \begin{array}{ccc}a& {a}^{2} &1\\b& {b}^{2} &1\\c& {c}^{2} &1\end{array}\right | \end{gathered}$

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}a& {a}^{2} &1\\b& {b}^{2} &1\\c& {c}^{2} &1\end{array}\right | \end{gathered}$

$\boxed{\sf{  \:OP \: C_2 \: \leftrightarrow \: C_3 \: \: }}$

$\rm \: = \: - \: \begin{gathered}\sf \left | \begin{array}{ccc}a& 1 & {a}^{2} \\b& 1 & {b}^{2} \\c&1 & {c}^{2} \end{array}\right | \end{gathered}$

$\boxed{\sf{  \:OP \: C_2 \: \leftrightarrow \: C_1 \: \: }}$

$\rm \: = \: (- )( - 1) \: \begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\1&b& {b}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered}$

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\1&b& {b}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered}$

Hence,

$\rm\implies \:\boxed{\sf{  \:\begin{gathered}\sf \left | \begin{array}{ccc}1&a&bc\\1&b&ca\\1&c& ab\end{array}\right | \end{gathered} = \begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\1&b& {b}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered} \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. The determinant value remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.

Answer 2

solution :

• please check the attached file