If A- a b c- b c a- c a b -abc- A T A-1- then find the value of a3-b3-c3-

A=[ a amp; b amp; c; b amp; c amp; a; c amp; a amp; b ], A ^ T =[ a amp; b amp; c; b amp; c amp; a; c amp; a amp; b ] given ...

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Byju's Answer

Standard XII

Mathematics

Singular and Non Singualar Matrices

If A=[ a b ...

Question

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

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦, A^{T} = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$
given $A A^{T} = I ∴ ∣ ∣ A A^{T} ∣ ∣ = | I | = 1$
or ${∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣}^{2} = 1 o r ∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = \pm 1$
or $3 a b c - a^{3} - b^{3} - c^{3} = \pm 1$
or $3 \mp 1 = a^{3} + b^{3} + c^{3}$
$∴$ $a^{3} + b^{3} + c^{3} = 2$ or $4$

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If A=⎡⎢⎣abcbcacab⎤⎥⎦,abc=ATA=1, then find the value of a3+b3+c3.

A=⎡⎢⎣abcbcacab⎤⎥⎦,AT=⎡⎢⎣abcbcacab⎤⎥⎦given AAT=I∴∣∣AAT∣∣=|I|=1or ∣∣ ∣∣abcbcacab∣∣ ∣∣2=1or∣∣ ∣∣abcbcacab∣∣ ∣∣=±1or 3abc−a3−b3−c3=±1or 3∓1=a3+b3+c3∴ a3+b3+c3=2 or 4

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