The value of the determinant 111 mC1 m+1C1 m+2C1 mC2 m+1C2 m+2C2 is equal to

111 mC1 m+1C1 m+2C1 mC2 m+1C2 m+2C2 =111 mC1 m+1C1 m+1C0+m+1C1 mC2 m+1C2 m+1C1+m+1C2=110 mC1 m+1C1 m+1C0 mC2 m+1C2 m+1C1 Applying C3→C3−C2=110 mC1 mC0+mC1 m+1C0 mC2 mC1+mC2 m+1C1=100 mC1 mC0 m+1C0 mC2 mC1 m+1C1 [Applying C2→C2−C1] =mC0m+1C1−m+1C0mC1=m+1−m=1

The value of the determinant 111 mC1 m+1C1 m+2C1 mC2 m+1C2 m+2C2 is equal to

111 mC1 m+1C1 m+2C1 mC2 m+1C2 m+2C2 =111 mC1 m+1C1 m+1C0+m+1C1 mC2 m+1C2 m+1C1+m+1C2=110 mC1 m+1C1 m+1C0 mC2 m+1C2 m+1C1 Applying C3→C3−C2=110 mC1 mC0+mC1 m+1C0 mC2 mC1+mC2 m+1C1=100 mC1 mC0 m+1C0 mC2 mC1 m+1C1 [Applying C2→C2−C1] =mC0m+1C1−m+1C0mC1=m+1−m=1

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

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Detailed Solution

$|\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 2} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 2} C_{2} \end{array}|$

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