Question

let a, b, c € R be all non - zero and satisfy a³+b³+c³ = 2if the matrix A = [a b c ]
[b c a ]
[ c a b ]
satisfies A transpose ×A = I then the value of ABC can be​

Answer 1

Given : a, b, c € R be all non - zero and satisfy a³+b³+c³ = 2

$A=\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$

A' * A = I

A'   is A transpose

To Find :  value of  a, b and c

Solution:

$A=\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$

$A^T=\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$

A' * A   = I  

=> a² + b²  + c²   = 1

    ab + bc + ac  = 0

a² + b²  + c²   = 1  

a, b, c € R be all non - zero

=> a³+b³+c³  < 1

but given a³+b³+c³  =   2

Hence no ( a , b , c) exist satisfying this

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Answer 2

Step-by-step explanation:

Solution:

\begin{gathered}A=\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]\end{gathered}

A=

⎣

⎢

⎡

a

b

c

b

c

a

c

a

b

⎦

⎥

⎤

\begin{gathered}A^T=\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]\end{gathered}

A

T

=

⎣

⎢

⎡

a

b

c

b

c

a

c

a

b

⎦

⎥

⎤

A' * A = I

=> a² + b² + c² = 1

ab + bc + ac = 0

a² + b² + c² = 1

a, b, c € R be all non - zero

=> a³+b³+c³ < 1

but given a³+b³+c³ = 2