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

Solution

$|\begin{matrix} - a^{2} & ab & ac \\ ba & - b^{2} & bc \\ ca & cb & - c^{2} \end{matrix}| = abc |\begin{matrix} - a & b & c \\ a & - b^{} & c \\ a & b & - c \end{matrix}|$

[ Taking out a, b, and c common from R₁, R₂, and R₃ respectively]

$= a^{2} b^{2} c^{2} |\begin{matrix} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix}|$

[ Taking out a, b, and c common from C₁, C₂, and C₃ respectively]

$= a^{2} b^{2} c^{2} |\begin{matrix} - 1 & 1 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}| . . . . . . . . . . [Applying R_{2} \to R_{2} + R_{1} and R_{3} \to R_{3} + R_{1}]$

$= a^{2} b^{2} c^{2} [(- 1) (0 x 0 - 2 x 2)] = a^{2} b^{2} c^{2} [- (0 - 4)] = 4 a^{2} b^{2} c^{2}$

Hence proved.

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Determinants

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

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Determinants

Using properties of determinants, prove that - a2 ab ac ba -b2 bc ca cb - c2 = 4 a2b2c2

Some More Questions From Determinants Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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