Find the inverse of a 11 = -3 a22 = -3 a33 = -3 a12 = a13 = a21 - Maths - Matrices

Question

Find the inverse of

a 11 = -3
a22 = -3
a33 = -3
a12 = a13 = a21 =
a23 = a31 = a32 = 2
-3 2 2
2 -3 2
2 2 -3

Harleen Birdi · Accepted Answer

$Let A = [\begin{matrix} - 3 & 2 & 2 \\ 2 & - 3 & 2 \\ 2 & 2 & - 3 \end{matrix}] | A | = [\begin{matrix} - 3 & 2 & 2 \\ 2 & - 3 & 2 \\ 2 & 2 & - 3 \end{matrix}] = - 3 (9 - 4) - 2 (- 6 - 4) + 2 (4 + 6) = 25 \neq 0$

Let aj be the cofactors of elements aij in A = [aij].

Then

$C_{11} = 5 C_{12} = 10 C_{13} = 10 C_{21} = 10 C_{22} = 5 C_{23} = 10 C_{31} = 10 C_{32} = 10 C_{33} = 5$

$∴ Adj A = [\begin{matrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{matrix}] = [\begin{matrix} 5 & 10 & 10 \\ 10 & 5 & 10 \\ 10 & 10 & 5 \end{matrix}] ∴ A^{- 1} = \frac{adj A}{| A |} = \frac{1}{25} [\begin{matrix} 5 & 10 & 10 \\ 10 & 5 & 10 \\ 10 & 10 & 5 \end{matrix}] = [\begin{matrix} \frac{1}{5} & \frac{2}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{2}{5} & \frac{1}{5} \end{matrix}]$

Find the inverse of a 11 = -3a22 = -3a33 = -3 a12 = a13 = a21 = a23 = a31 = a32 = 2 -3 2 22 -3 22 2 -3

Find the inverse of

a ₁₁ = -3
a₂₂ = -3
a₃₃ = -3

a₁₂ = a₁₃ = a₂₁ = a₂₃ = a₃₁ = a₃₂ = 2

-3 2 2
2 -3 2
2 2 -3