Determinants & Cramer's Rule

A determinant is a scalar value that is a function of the entries of a square matrix. It provides important information about the matrix, such as whether it is invertible.

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is calculated as \( det(A) = ad - bc \).

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.

Cramer's Rule can be applied when the system of equations has a unique solution, which occurs if the determinant of the coefficient matrix is non-zero.

For a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \), the determinant is calculated as \( det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

If the determinant of a matrix is zero, it indicates that the matrix is singular, meaning it does not have an inverse and the system of equations may have either no solution or infinitely many solutions.

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the inverse is given by \( A^{-1} = \frac{1}{det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \) if \( det(A) \neq 0 \).

The determinant of a matrix representing a linear transformation indicates the scaling factor of the transformation and whether it preserves orientation.

The area of a triangle formed by points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) can be found using the determinant: \( Area = \frac{1}{2} \left| det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right| \).

The cofactor of an element \( a_{ij} \) in a matrix is given by \( C_{ij} = (-1)^{i+j} det(M_{ij}) \), where \( M_{ij} \) is the submatrix formed by deleting the i-th row and j-th column.