Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems If Δ = ∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} Δ = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33
Mathematics
Problem If Δ = ∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} Δ = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 C i j = ( − 1 ) i + j M i j C_{ij} = (-1)^{i+j} M_{ij} C ij = ( − 1 ) i + j M ij M i j M_{ij} M ij i i i j j j ∣ C 11 C 12 C 13 C 21 C 22 C 23 C 31 C 32 C 33 ∣ = Δ 2 \begin{vmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{vmatrix} = \Delta^2 C 11 C 21 C 31 C 12 C 22 C 32 C 13 C 23 C 33 = Δ 2 ∣ 1 x x 2 x x 2 1 x 2 1 x ∣ = 5 \begin{vmatrix} 1 & x & x^2 \\ x & x^2 & 1 \\ x^2 & 1 & x \end{vmatrix} = 5 1 x x 2 x x 2 1 x 2 1 x = 5 Δ = ∣ x 3 − 1 0 x − x 4 0 x − x 4 x 3 − 1 x − x 4 x 3 − 1 0 ∣ \Delta = \begin{vmatrix} x^3 - 1 & 0 & x - x^4 \\ 0 & x - x^4 & x^3 - 1 \\ x - x^4 & x^3 - 1 & 0 \end{vmatrix} Δ = x 3 − 1 0 x − x 4 0 x − x 4 x 3 − 1 x − x 4 x 3 − 1 0 Δ 2 \Delta^2 Δ 2 Browse the problem library