Question

Let matrix A =$\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ where a, b, c are real positive numbers with abc = 1. If A^T A = I, then a³ + b³ + c³ =

• A. 3
• B. 2
• C. 4
• D. None

Answer 1

Answer: (C)

4

Answer 2

A^TA = $\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ $\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$

= $\begin{bmatrix} a^{2} + b^{2} + c^{2} & ab + bc + ca & ac + ab + bc \\ ab + bc + ca & b^{2} + c^{2} + a^{2} & bc + ca + ab \\ ca + ab + bc & bc + ac + ab & c^{2} + a^{2} + b^{2} \end{bmatrix}$

A^TA = I Ž a² + b² + c² = 1, ab + bc + ca = 0

\ (a + b + c)² = 1 + 2 (0) = 1 Ž a + b + c = 1

a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ca)

a³ + b³ + c² = 3x 1 + 1 = 4