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Question: Let $\Delta$=$\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\ c^2 & 2ca-b^2 & a^2 \\ b^2 & a^2 & 2ab-c^2 \end...

Let Δ\Delta=∣2bc−a2c2b2c22ca−b2a2b2a22ab−c2∣\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\ c^2 & 2ca-b^2 & a^2 \\ b^2 & a^2 & 2ab-c^2 \end{vmatrix}, then Δ\Delta can be expressed as

A

∣abcbcacab∣2\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}^2

B

∣cbaacbbac∣2\begin{vmatrix} c & b & a \\ a & c & b \\ b & a & c \end{vmatrix}^2

C

∣cabcbacab∣2\begin{vmatrix} c & a & b \\ c & b & a \\ c & a & b \end{vmatrix}^2

D

None

Answer

∣abcbcacab∣2\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}^2

Explanation

Solution

Let M=(abcbcacab)M = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}. Then det⁡(M)=a(bc−a2)−b(b2−ac)+c(ab−c2)=3abc−a3−b3−c3\det(M) = a(bc - a^2) - b(b^2 - ac) + c(ab - c^2) = 3abc - a^3 - b^3 - c^3.

Let N=(−acbc−baba−c)N = \begin{pmatrix} -a & c & b \\ c & -b & a \\ b & a & -c \end{pmatrix}. Then det⁡(N)=−a(bc−a2)−c(−c2−ab)+b(ac+b2)=−abc+a3+c3+abc+abc+b3=a3+b3+c3+3abc\det(N) = -a(bc - a^2) - c(-c^2 - ab) + b(ac + b^2) = -abc + a^3 + c^3 + abc + abc + b^3 = a^3 + b^3 + c^3 + 3abc.

MN=(−a2+b2+c2ab−bc+caac+ab−bc−ab+bc+acb2−c2+a2bc+ca−ab−ac+ab+bcbc−ac+abc2+a2−b2)MN = \begin{pmatrix} -a^2+b^2+c^2 & ab-bc+ca & ac+ab-bc \\ -ab+bc+ac & b^2-c^2+a^2 & bc+ca-ab \\ -ac+ab+bc & bc-ac+ab & c^2+a^2-b^2 \end{pmatrix}

Consider (abcbcacab)(−acbc−baba−c)=(−a2+bc+bcac−b2+acab+ab−c2−ab+c2+abbc−bc+a2b2+ac−ac−ac+ac+b2c2−ab+abbc+a2−bc)=(2bc−a22ac−b22ab−c2c2a2b2b2c2a2)\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \begin{pmatrix} -a & c & b \\ c & -b & a \\ b & a & -c \end{pmatrix} = \begin{pmatrix} -a^2+bc+bc & ac-b^2+ac & ab+ab-c^2 \\ -ab+c^2+ab & bc-bc+a^2 & b^2+ac-ac \\ -ac+ac+b^2 & c^2-ab+ab & bc+a^2-bc \end{pmatrix} = \begin{pmatrix} 2bc-a^2 & 2ac-b^2 & 2ab-c^2 \\ c^2 & a^2 & b^2 \\ b^2 & c^2 & a^2 \end{pmatrix}

Δ=∣2bc−a2c2b2c22ca−b2a2b2a22ab−c2∣\Delta = \begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\ c^2 & 2ca-b^2 & a^2 \\ b^2 & a^2 & 2ab-c^2 \end{vmatrix}

det⁡(M)det⁡(N)=(3abc−a3−b3−c3)(3abc+a3+b3+c3)=(3abc)2−(a3+b3+c3)2\det(M) \det(N) = (3abc - a^3 - b^3 - c^3)(3abc + a^3 + b^3 + c^3) = (3abc)^2 - (a^3 + b^3 + c^3)^2. det⁡(M)2=(3abc−a3−b3−c3)2=(a3+b3+c3−3abc)2\det(M)^2 = (3abc - a^3 - b^3 - c^3)^2 = (a^3 + b^3 + c^3 - 3abc)^2.

The determinant of the given matrix is (a3+b3+c3−3abc)2(a^3 + b^3 + c^3 - 3abc)^2.