Determinants

Algebra Level 4

∣ x 2 + x x + 1 x − 2 2 x 2 + 3 x − 1 3 x 3 x − 3 x 2 + 2 x + 3 2 x − 1 2 x − 1 ∣ = A x + B \left| \begin{matrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{matrix} \right| = Ax+B

A A and B B above are determinants of order 3 3 . Find the value of A + 2 B A+2B .

3 3 None of these \text{ None of these } 1 1 0 0 2 2

Sabhrant Sachan by Sabhrant Sachan , 21, India

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1 solution

Sabhrant Sachan
Nov 21, 2016

C = ∣ x 2 + x x + 1 x − 2 2 x 2 + 3 x − 1 3 x 3 x − 3 x 2 + 2 x + 3 2 x − 1 2 x − 1 ∣ C = ∣ x 2 + x x + 1 x − 2 x − 1 x − 2 x + 1 x + 3 x − 2 x + 1 ∣ i ) R 2 → R 2 − 2 R 1 i i ) R 3 → R 3 − R 1 C = ∣ x 2 + x x + 1 x − 2 − 4 0 0 x + 3 x − 2 x + 1 ∣ i ) R 2 → R 2 − R 3 i i ) Open det from R 2 C = 4 [ ( x + 1 ) 2 − ( x − 2 ) 2 ] C = \left| \begin{matrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{matrix} \right| \\ C = \left| \begin{matrix} x^2+x & x+1 & x-2 \\ x-1 & x-2 & x+1 \\ x+3 & x-2 & x+1 \end{matrix} \right| \quad \color{#3D99F6}{\begin{matrix} \small{i) R_2\rightarrow R_2-2R_1} \\ \small{ii) R_3\rightarrow R_3-R_1} \end{matrix} } \\ C = \left| \begin{matrix} x^2+x & x+1 & x-2 \\ -4 & 0 & 0 \\ x+3 & x-2 & x+1 \end{matrix} \right| \quad \color{#3D99F6}{\begin{matrix} \small{i) R_2\rightarrow R_2-R_3} \\ \small{ii) \text{ Open det from } R_2 } \end{matrix} } \\ C = 4[(x+1)^2-(x-2)^2]

C = 24 x − 12 = A x + B A = 24 , B = − 12 A + 2 B = 0 C = 24x-12 = Ax+B \\ A=24 \text{ , }B=-12 \\ \boxed{A+2B = 0}

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