SATHEE: Matrices and Determinants 2 Question 34

Matrices and Determinants 2 Question 34

38. Let $a > 0, d > 0$ . Find the value of the determinant

$| \begin{matrix} \frac{1}{a} & \frac{1}{a (a + d)} & \frac{1}{(a + d) (a + 2 d)} \\ \frac{1}{(a + d)} & \frac{1}{(a + d) (a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} \\ \frac{1}{(a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} & \frac{1}{(a + 3 d) (a + 4 d)} \end{matrix} |$

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Answer:

Correct Answer: 38.

$\frac{4 d^{4}}{a (a + d)^{2} (a + 2 d)^{3} (a + 3 d)^{2} (a + 4 d)}$

Solution:

Given, $a > 0, d > 0$ and let

$Δ = | \begin{matrix} \frac{1}{a} & \frac{1}{a (a + d)} & \frac{1}{(a + d) (a + 2 d)} \\ \frac{1}{(a + d)} & \frac{1}{(a + d) (a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} \\ \frac{1}{(a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} & \frac{1}{(a + 3 d) (a + 4 d)} \end{matrix} |$

Taking $\frac{1}{a (a + d) (a + 2 d)}$ common from $R_{1}$ ,

$\begin{aligned} \frac{1}{(a + d) (a + 2 d) (a + 3 d)} from R_{2}, \\ \frac{1}{(a + 2 d) (a + 3 d) (a + 4 d)} from R_{3} \\ \Rightarrow Δ = \frac{1}{a (a + d)^{2} (a + 2 d)^{3} (a + 3 d)^{2} (a + 4 d)} \\ (a + d) (a + 2 d) (a + 2 d) a \\ (a + 2 d) (a + 3 d) (a + 3 d) (a + d) \\ (a + 3 d) (a + 4 d) (a + 4 d) (a + 2 d) \end{aligned}$

$\Rightarrow Δ = \frac{1}{a (a + d)^{2} (a + 2 d)^{3} (a + 3 d)^{2} (a + 4 d)} Δ^{'}$

where, $Δ^{'} = | \begin{matrix} (a + d) (a + 2 d) & (a + 2 d) & a \\ (a + 2 d) (a + 3 d) & (a + 3 d) & (a + d) \\ (a + 3 d) (a + 4 d) & (a + 4 d) & (a + 2 d) \end{matrix} |$

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

$\Rightarrow Δ^{'} = | \begin{matrix} (a + d) (a + 2 d) & (a + 2 d) & a \\ (a + 2 d) (2 d) & d & d \\ (a + 3 d) (2 d) & d & d \end{matrix} |$

Applying $R_{3} \to R_{3} - R_{2}$

$Δ^{'} = | \begin{matrix} (a + d) (a + 2 d) & (a + 2 d) & a \\ (a + 2 d) 2 d & d & d \\ 2 d^{2} & 0 & 0 \end{matrix} |$

Expanding along $R_{3}$ , we get

$\begin{aligned} Δ^{'} = 2 d^{2} | \begin{array}{c} a + 2 d & a \\ d & d \end{array} | \\ Δ^{'} = (2 d^{2}) (d) (a + 2 d - a) = 4 d^{4} \\ ∴ Δ = \frac{4 d^{4}}{a (a + d)^{2} (a + 2 d)^{3} (a + 3 d)^{2} (a + 4 d)} \end{aligned}$

Matrices and Determinants 2 Question 34

38. Let $a > 0, d > 0$ . Find the value of the determinant

Answer:

Solution:

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Matrices and Determinants 2 Question 34

38. Let a>0,d>0. Find the value of the determinant

Answer:

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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38. Let $a > 0, d > 0$ . Find the value of the determinant