First slide

Applications of determinant

Question

$Let a determinant \underset{}{\underset{}{be given}} by A = |\begin{array}{lll} a b c \\ p q r \\ x y z \end{array}| and suppose that det. A = 6 If$ $B = |\begin{array}{lll} p + x q + y r + z \\ a + x b + y c + z \\ a + p b + q c + r \end{array}| then \det (B) is$

Moderate

Solution

$We have B = |\begin{array}{lll} p + x q + y r + z \\ a + x b + y c + z \\ a + p b + q c + r \end{array}|$
$Apply R_{1} \to R_{1} + R_{2} + R_{3}, we get$
$B = 2 |\begin{array}{ccc} a + p + x & b + q + y & c + r + z \\ a + x & b + y & c + z \\ a + p & b + q & c + r \end{array}|$
$Apply R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1}, we get$
$= 2 |\begin{matrix} a + p + x & b + q + y & c + r + z \\ - p & - q & - r \\ - x & - y & - z \end{matrix}|$
$Apply R_{1} \to R_{1} + R_{2} + R_{3}$
$∴ det. B = 2 (6) = 12$

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