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Adjoint of a Matrix

If Tm,Tn,Tk...

Question

If $T_{m}, T_{n}, T_{k}$ are $m^{t h}, n^{t h}, & k^{t h}$ terms of an AP then prove that $∣ ∣ ∣ ∣ \begin{matrix} T_{m} & m & 1 T_{n} & n & 1 T_{k} & k & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

Let the first term of $A P$ be $a$ and common difference be $d$

$⟹ T_{m} = a + (m - 1) d, T_{n} = a + (n - 1) d, T_{k} = a + (k - 1) d$

$∣ ∣ ∣ ∣ \begin{matrix} T_{m} & m & 1 T_{n} & n & 1 T_{k} & k & 1 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + (m - 1) d & m & 1 a + (n - 1) d & n & 1 a + (k - 1) d & k & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}$

$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} (m - n) d & (m - n) & 0 (n - k) d & (n - k) & 0 a + (k - 1) d & k & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = (m - n) (n - k) ∣ ∣ ∣ ∣ \begin{matrix} d & 1 & 0 d & 1 & 0 a + (k - 1) d & k & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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