SATHEE: Matrices and Determinants 1 Question 20

Matrices and Determinants 1 Question 20

Analytical and Descriptive Questions

20. If matrix $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ , where $a, b, c$ are real positive numbers, $a b c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .

$(2003, 2 M)$

Integer Type Question

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Answer:

Correct Answer: 20. (4)

Solution:

Given, $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}], a b c = 1$ and $A^{T} A = I$ …(i)

Now, $A^{T} A = I$

$[\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ $[\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ $= [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

$[\begin{matrix} a^{2} + & b^{2} + & c^{2} \\ a b + & b c + & c a \\ a b + & b c + & c a \end{matrix}]$ $[\begin{matrix} a b + & b c + & c a \\ a^{2} + & b^{2} + & c^{2} \\ a b + & b c + & c a \end{matrix}]$ $[\begin{matrix} a b + & b c + & c a \\ a b + & b c + & c a \\ a^{2} + & b^{2} + & c^{2} \end{matrix}]$

$= [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

$\Rightarrow a^{2} + b^{2} + c^{2} = 1$ and $a b + b c + c a = 0$ …(ii)

We know, $a^{3} + b^{3} + c^{3} - 3 a b c$

$= (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$

$\Rightarrow a^{3} + b^{3} + c^{3} = (a + b + c) (1 - 0) + 3$

[from Eqs. (i) and (ii)]

$∴ a^{3} + b^{3} + c^{3} = (a + b + c) + 3$ …(iii)

Now, $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$

$= 1$

From Eq. (iii), $a^{3} + b^{3} + c^{3} = 1 + 3 \Rightarrow a^{3} + b^{3} + c^{3} = 4$

Matrices and Determinants 1 Question 20

20. If matrix $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ , where $a, b, c$ are real positive numbers, $a b c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .

Integer Type Question

Answer:

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Matrices and Determinants 1 Question 20

20. If matrix A=[abcbcacab], where a,b,c are real positive numbers, abc=1 and ATA=I, then find the value of a3+b3+c3.

Integer Type Question

Answer:

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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20. If matrix $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ , where $a, b, c$ are real positive numbers, $a b c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .