The determinant for a 2 × 2 matrix is defined as

⎢ ⎢ ⎢ ⎢ a11 a12 a21 a22 ⎢ ⎢ ⎢ ⎢ ≡ a11 a22 − a12 a21.

Using this notation, the solution to the simultaneous equations,

a11 x1 + a12 x2 = b1 a21 x1 + a22 x2 = b2,

can be written as

x1 = ⎢ ⎢ ⎢ ⎢ b1 a12 b2 a22 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ a11 a12 a21 a22 ⎢ ⎢ ⎢ ⎢ x2 = ⎢ ⎢ ⎢ ⎢ a11 b1 a12 b2 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ a11 a12 a21 a22 ⎢ ⎢ ⎢ ⎢ ,

which is the famous Cramer's rule.

Three simultaneous equations can be solved in a similar manner as

⎧ ⎪ ⎨ ⎪ ⎩ a11 x1 + a12 x2 + a13x3 = b1 a21 x1 + a22 x2 + a23x3 = b2 a31 x1 + a32 x2 + a33x3 = b3,

(1)

x1 = ⎢ ⎢ ⎢ ⎢ ⎢ b1 a12 a13 b2 a22 a23 b3 a32 a33 ⎢ ⎢ ⎢ ⎢ ⎢ D ,   x2 = ⎢ ⎢ ⎢ ⎢ ⎢ a11 b1 a13 a21 b2 a23 a31 b3 a33 ⎢ ⎢ ⎢ ⎢ ⎢ D ,   x3 = ⎢ ⎢ ⎢ ⎢ ⎢ a11 a12 b1 a21 a22 b2 a31 a32 b3 ⎢ ⎢ ⎢ ⎢ ⎢ D .

where

D ≡ ⎢ ⎢ ⎢ ⎢ ⎢ a11 a12 a13 a21 a22 a23 a31 a32 a33 ⎢ ⎢ ⎢ ⎢ ⎢ ,

and the determinant for 3 × 3 matrices is defined as

⎢ ⎢ ⎢ ⎢ ⎢ a11 a12 a13 a21 a22 a23 a31 a32 a33 ⎢ ⎢ ⎢ ⎢ ⎢ = a11 a22 a33 + a12 a23 a31 + a21 a32 a13 −a13 a22 a31 − a12 a21 a33 − a23 a32 a11.

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File translated from T_EX by T_TH, version 4.03.
On 25 Jun 2023, 21:38.