First slide

Introduction to Determinants

Question

If A₁, B₁, C₁, . . . are, respectively, the cofactors of the elements a₁, b₁, c₁, . . . of the determinant $Δ = |\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}|, Δ \neq 0$ , then the value of $|\begin{array}{ll} B_{2} C_{2} \\ B_{3} C_{3} \end{array}|$ is equal to

Difficult

A

$a_{1}^{2} Δ$
B

$a_{1} Δ$
C

$a_{1} Δ^{2}$
D

$a_{1}^{2} Δ^{2}$

Solution

$\begin{aligned} B_{2} = a_{1} c_{3} - a_{3} c_{1}, C_{2} = - (a_{1} b_{3} - a_{3} b_{1}) \\ B_{3} = - (a_{1} c_{2} - a_{2} c_{1}), C_{3} = a_{1} b_{2} - a_{2} b_{1} \end{aligned}$
$\begin{matrix} ∴ |\begin{array}{ll} B_{2} & C_{2} \\ B_{3} & C_{3} \end{array}| = |\begin{array}{cc} a_{1} c_{3} - a_{3} c_{1} & - a_{1} b_{3} + a_{3} b_{1} \\ - a_{1} c_{2} + a_{2} c_{1} & a_{1} b_{2} - a_{2} b_{1} \end{array}| \\ = |\begin{array}{cc} a_{1} c_{3} & - a_{1} b_{3} \\ - a_{1} c_{2} & a_{1} b_{2} \end{array}| + |\begin{array}{cc} a_{1} c_{3} & a_{3} b_{1} \\ - a_{1} c_{2} & - a_{2} b_{1} \end{array}| + |\begin{array}{cc} - a_{3} c_{1} & - a_{1} b_{3} \\ a_{2} c_{1} & a_{1} b_{2} \end{array}| + |\begin{array}{cc} - a_{3} c_{1} & a_{3} b_{1} \\ a_{2} c_{1} & - a_{2} b_{1} \end{array}| \end{matrix}$
$= a_{1}^{2} |\begin{array}{cc} c_{3} & - b_{3} \\ - c_{2} & b_{2} \end{array}| + a_{1} b_{1} |\begin{array}{cc} c_{3} & a_{3} \\ - c_{2} & - a_{2} \end{array}| + a_{1} c_{1} |\begin{array}{cc} - a_{3} & - b_{3} \\ a_{2} & b_{2} \end{array}| + b_{1} c_{1} |\begin{array}{cc} - a_{3} & a_{3} \\ a_{2} & - a_{2} \end{array}|$
$= a_{1} \{a_{1} (b_{2} c_{3} - b_{3} c_{2}) - b_{1} (a_{2} c_{3} - a_{3} c_{2}) + c_{1} (a_{2} b_{3} - a_{3} b_{2})\}$
$= a_{1} |\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}| = a_{1} Δ$

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