If A=0 c −b−c 0 ab −a 0 then A3 is

A2=0c−b−c0ab−a00c−b−c0ab−a0=0−c2−b20+0+ab0+ac+00+0+ab−c2+0−a2bc+0+00+ac+0bc+0+0−b2−a2+0=−b2−c2 ab acab −c2−a2bcac bc −a2−b2A3=0c−b−c0ab−a0−b2−c2abacab−c2−a2bcacbc−a2−b2=0+abc−abc0−c3−a2c−b2c0+bc2+a2b+b3b2c+c3+0+a2c−abc+0+abc−ac2+0−a3−ab2−b3−bc2−a2b+0ab2+ac2+a3+00=0−ca2+b2+c2ba2+b2+c2ca2+b2+b20−aa2+b2+c2−ba2+b2+c2aa2+b2+c20=−a2+b2+c20c−b−c0ab−a0=−a2+b2+c2A. ∴A3+a2+b2+c2A=O

If A=0 c −b−c 0 ab −a 0 then A3 is

A2=0c−b−c0ab−a00c−b−c0ab−a0=0−c2−b20+0+ab0+ac+00+0+ab−c2+0−a2bc+0+00+ac+0bc+0+0−b2−a2+0=−b2−c2 ab acab −c2−a2bcac bc −a2−b2A3=0c−b−c0ab−a0−b2−c2abacab−c2−a2bcacbc−a2−b2=0+abc−abc0−c3−a2c−b2c0+bc2+a2b+b3b2c+c3+0+a2c−abc+0+abc−ac2+0−a3−ab2−b3−bc2−a2b+0ab2+ac2+a3+00=0−ca2+b2+c2ba2+b2+c2ca2+b2+b20−aa2+b2+c2−ba2+b2+c2aa2+b2+c20=−a2+b2+c20c−b−c0ab−a0=−a2+b2+c2A. ∴A3+a2+b2+c2A=O

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If $A = [\begin{array}{rrr} 0 c - b \\ - c 0 a \\ b - a 0 \end{array}]$ then A³ is

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$(a^{2} + b^{2} + c^{2}) A$

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

answer is D.

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Detailed Solution

$\begin{array}{l} A^{2} = [\begin{array}{rrr} 0 & c & - b \\ - c & 0 & a \\ b & - a & 0 \end{array}] [\begin{array}{rrr} 0 & c & - b \\ - c & 0 & a \\ b & - a & 0 \end{array}] = [\begin{array}{ccc} 0 - c^{2} - b^{2} & 0 + 0 + a b & 0 + a c + 0 \\ 0 + 0 + a b & - c^{2} + 0 - a^{2} & b c + 0 + 0 \\ 0 + a c + 0 & b c + 0 + 0 & - b^{2} - a^{2} + 0 \end{array}] \\ = [\begin{array}{rrc} - b^{2} - c^{2} & a b & a c \\ a b & - c^{2} - a^{2} & b c \\ a c & b c & - a^{2} - b^{2} \end{array}] \\ A^{3} = [\begin{array}{rrr} 0 & c & - b \\ - c & 0 & a \\ b & - a & 0 \end{array}] [\begin{array}{ccc} - b^{2} - c^{2} & a b & a c \\ a b & - c^{2} - a^{2} & b c \\ a c & b c & - a^{2} - b^{2} \end{array}] \end{array}$
$\begin{array}{l} = [\begin{array}{ccc} 0 + a b c - a b c & 0 - c^{3} - a^{2} c - b^{2} c & 0 + b c^{2} + a^{2} b + b^{3} \\ b^{2} c + c^{3} + 0 + a^{2} c & - a b c + 0 + a b c & - a c^{2} + 0 - a^{3} - a b^{2} \\ - b^{3} - b c^{2} - a^{2} b + 0 & a b^{2} + a c^{2} + a^{3} + 0 & 0 \end{array}] \\ = [\begin{array}{ccc} 0 & - c (a^{2} + b^{2} + c^{2}) & b (a^{2} + b^{2} + c^{2}) \\ c (a^{2} + b^{2} + b^{2}) & 0 & - a (a^{2} + b^{2} + c^{2}) \\ - b (a^{2} + b^{2} + c^{2}) & a (a^{2} + b^{2} + c^{2}) & 0 \end{array}] \\ = - (a^{2} + b^{2} + c^{2}) [\begin{array}{rrr} 0 & c & - b \\ - c & 0 & a \\ b & - a & 0 \end{array}] = - (a^{2} + b^{2} + c^{2}) A . ∴ A^{3} + (a^{2} + b^{2} + c^{2}) A = O \end{array}$