Find the determinant of the matrix

Algebra Level 2

Find the determinant of matrix A A .

A = [ a b a c a b b c c ] \large{A= \begin{bmatrix} a & b & a \\ c & a & b \\ b & c & c \end{bmatrix}}

a 2 ( c − b ) + b 3 − a c + ( a − b ) c 2 a^2(c-b)+b^3-ac+(a-b)c^2 b 3 + a 2 c − b ( a c − c 2 ) + a c 2 b^3+a^2c-b(ac-c^2)+ac^2 a c 2 + b 3 − b c 2 + a 2 c − a b + a c b ac^2+b^3-bc^2+a^2c-a^b+acb b ( b 2 − a c ) + a 2 ( c − b ) + c 2 ( a − b ) b(b^2-ac)+a^2(c-b)+c^2(a-b)

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1 solution

detA = ∣ a b a a b c a b c a b c c b c ∣ \large{\text{detA}= \begin{vmatrix} a & b & a & a & b\\ c & a & b & c & a\\ b & c & c & b & c \end{vmatrix}}

= [ a 2 c + b 3 + a c 2 − ( a 2 b + a c b + b c 2 ) ] = a 2 c + b 3 + a c 2 − a 2 b − a c b − b c 2 = c 2 ( a − b ) + a 2 ( c − b ) + b ( b 2 − a c ) \large{=\left[a^2c+b^3+ac^2-(a^2b+acb+bc^2)\right]=a^2c+b^3+ac^2-a^2b-acb-bc^2=c^2(a-b)+a^2(c-b)+b(b^2-ac)}

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