First slide

Introduction to Determinants

Question

The determinant $|\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 2} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 2} C_{2} \end{array}|$ value is equal to

Difficult

A

1
B

-1
C

0
D

none of these

Solution

$|\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 2} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 2} C_{2} \end{array}|$
$\begin{matrix} = |\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 1} C_{0} +^{m + 1} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 1} C_{1} +^{m + 1} C_{2} \end{array}| \\ = |\begin{array}{ccc} 1 & 1 & 0 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 1} C_{1} \end{array}| [Applying C_{3} \to C_{3} - C_{2}] \\ = |\begin{array}{ccc} 1 & 1 & 0 \\ ^{m} C_{1} & ^{m} C_{0} +^{m} C_{1} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m} C_{1} +^{m} C_{2} & ^{m + 1} C_{1} \end{array}| \\ = |\begin{array}{ccc} 1 & 0 & 0 \\ ^{m} C_{1} & ^{m} C_{0} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m} C_{1} & ^{m + 1} C_{1} \end{array}| [Applying C_{2} \to C_{2} - C_{1}] \\ =^{m} {C_{0}}^{m + 1} C_{1} -^{m + 1} {C_{0}}^{m} C_{1} \\ = m + 1 - m \\ = 1 \end{matrix}$

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