SATHEE: Matrices and Determinants 2 Question 43

Matrices and Determinants 2 Question 43

47. Let $a, b, c$ be positive and not all equal. Show that the value of the determinant

$| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} |$ is negative.

$(1981, 4 M)$

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Solution:

Let $Δ = | \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} |$

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$

$Δ = | \begin{matrix} a + b + c & b & c \\ a + b + c & c & a \\ a + b + c & a & b \end{matrix} | = (a + b + c) | \begin{matrix} 1 & b & c \\ 1 & c & a \\ 1 & a & b \end{matrix} |$

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

$\begin{aligned} = (a + b + c) | \begin{array}{c} 1 & b & c \\ 0 & c - b & a - c \\ 0 & a - b & b - c \end{array} | \\ = (a + b + c) [- (c - b)^{2} - (a - b) (a - c)] \\ = - (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) \\ = - \frac{1}{2} (a + b + c) (2 a^{2} + 2 b^{2} + 2 c^{2} - 2 a b - 2 b c - 2 c a) \\ = - \frac{1}{2} (a + b + c) [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}] \end{aligned}$

which is always negative.

Matrices and Determinants 2 Question 43

47. Let $a, b, c$ be positive and not all equal. Show that the value of the determinant

Solution:

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Matrices and Determinants 2 Question 43

47. Let a,b,c be positive and not all equal. Show that the value of the determinant

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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47. Let $a, b, c$ be positive and not all equal. Show that the value of the determinant