Given a matrix $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ , where a, b, c are real positive numbers, abc $=$ 1 and A $^{T}$ A $=$ I, then find the value of $a^{3} + b^{3} + c^{3}$ .

Open in App

Solution

Given $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$
Here, $A = A^{T}$
Also, $| A | = a (b c - a^{2}) - b (b^{2} - a c) + c (a b - c^{2})$
$\Rightarrow | A | = 3 a b c - a^{3} - b^{3} - c^{3}$
$\Rightarrow | A | = 3 - a^{3} - b^{3} - c^{3}$ ( $∵ a b c = 1$ given)
Also, given $A A^{T} = I$
$\Rightarrow A^{2} = I$
$\Rightarrow | A^{2} | = 1$
$\Rightarrow | A | = \pm 1$
$\Rightarrow 3 - a^{3} - b^{3} - c^{3} = - 1$ ( $∵$ a, b,c are positive real numbers and $3 a b c - a^{3} - b^{3} - c^{3} = - (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - a c)$ )
$\Rightarrow a^{3} + b^{3} + c^{3} = 4$

Suggest Corrections

Similar questions

If matrix $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ where a, b, c are real positive numbers, abc = 1 and $A^{T} A = I,$ then the value of $a^{3} + b^{3} + c^{3}$ is ___

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦, a b c = 1, A^{T} A = I

, then find the value of

a^{3} + b^{3} + c^{3}

Q. If matrix

A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦

where a,b,c are real numbers

A^{T} A = I

, then find the value of

a + b + c

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦, a b c = A^{T} A = 1

, then find the value of

a^{3} + b^{3} + c^{3}

Join BYJU'S Learning Program

Given a matrix A=⎡⎢⎣abcbcacab⎤⎥⎦, where a, b, c are real positive numbers, abc = 1 and ATA = I, then find the value of a3+b3+c3.

Given a matrix $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ , where a, b, c are real positive numbers, abc $=$ 1 and A $^{T}$ A $=$ I, then find the value of $a^{3} + b^{3} + c^{3}$ .