Summary

Determinant of a square matrix: For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det(A) = ad - bc$. For a $3 \times 3$ matrix, expand along any row/column using minors and cofactors.

Minors and cofactors: Minor $M_{ij}$ is determinant after removing row $i$ and column $j$. Cofactor $C_{ij} = (-1)^{i+j} M_{ij}$. Determinant = sum of (element $\times$ its cofactor) along any row/column.

Properties of determinants:

• Interchanging rows/columns changes sign.
• Two identical rows/columns $\Rightarrow$ determinant = 0.
• Multiplying a row by $k$ multiplies determinant by $k$.
• Adding a multiple of one row to another does not change determinant.

Area of a triangle: Area $= \frac{1}{2} \left| \det \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \right|$.

Adjoint and inverse: For a square matrix $A$, adjoint $(\text{adj } A)$ is transpose of cofactor matrix. $A^{-1} = \frac{1}{\det(A)} \text{adj } A$, provided $\det(A) \neq 0$.

Consistency of linear equations: For $AX = B$:

• If $\det(A) \neq 0$, unique solution (consistent).
• If $\det(A) = 0$ and $(\text{adj } A)B = 0$, infinite solutions (consistent).
• If $\det(A) = 0$ and $(\text{adj } A)B \neq 0$, no solution (inconsistent). …