Using properties of determinants, prove that -a^ 2 & ab & ac ba &… - Vidyadip

Question

Using properties of determinants, prove that $\begin{vmatrix} -\text{a}^{2} & \text{ab} & \text{ac} \\ \text{ba} & -\text{b}^{2} & \text{bc} \\ \text{ca} & \text{cb} & -\text{c}^{2} \end{vmatrix}=\text{4a}^{2}\text{b}^{2}\text{c}^{2} $.

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Answer

Taking $a, b, c$ respectively common from $R_1, R_2, R_3$_ to get
LHS = Determinant = abc $\begin{vmatrix} \text{-a} & \text{b} & \text{c} \\ \text{a} & \text{-b} & \text{c} \\ \text{a} & \text{b} & \text{-c} \end{vmatrix}$
TX
$LHS = a^2b^2c^2$ $\begin{vmatrix} \text{-1} & \text{1} & \text{1} \\ \text{1} & \text{-1} & \text{1} \\ \text{1} & \text{1} & \text{-1} \end{vmatrix}$
Applying $R_2$_$\rightarrow$$R_2+R_1, R_3$_ $\rightarrow$$R_3+ R_1,$ to get
$LHS = a^2b^2c^2 $$\begin{vmatrix} \text{-1} & \text{1} & \text{1} \\ \text{0} & \text{0} & \text{2} \\ \text{0} & \text{2} & \text{0} \end{vmatrix}$
$= a^2b^2c^{2 .}(-1) (-4) = 4 a^2b^2c^2= RHS.$

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