Definitions [2]
Definition: Determinant
A determinant is a number associated with a square matrix.
\[\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}=ad-bc\]
The value of the determinant is ad - bc.
The degree of a 2 × 2 determinant is 2.
Definition: Cramer’s Rule (Determinant Method)
Cramer’s Rule is a method to solve simultaneous linear equations using determinants.
-
It can be applied only when the determinant D ≠ 0
- Standard Form of Equations
a2x + b2y = c2
Formulae [1]
Formula: Determinant Method (Cramer’s Rule)
\[D=
\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix}=a_1b_2-a_2b_1\]
\[D_x=
\begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2
\end{vmatrix}=c_1b_2-c_2b_1\]
\[D_y=
\begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix}=a_1c_2-a_2c_1\]
\[x=\frac{D_x}{D}\quad\mathrm{and}\quad y=\frac{D_y}{D}\]
-
If D ≠ 0 → unique solution
-
If D = 0 → Cramer’s rule is not applicable
