Determinants and Cramer's Rule are powerful tools in linear algebra. They help us solve systems of equations and understand matrix properties. These concepts are crucial for grasping how matrices transform space and represent linear transformations.

Determinants give us a single number that tells us a lot about a matrix. Cramer's Rule uses determinants to solve equations. Together, they show how matrices, equations, and geometric transformations are all connected in cool ways.

Determinants of Square Matrices

Calculating Determinants

• The determinant is a scalar value computed from the elements of a square matrix
• For a 2x2 matrix, calculate the determinant by multiplying the main diagonal elements and subtracting the product of the other diagonal elements
  • Example: For matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $det(A) = ad - bc$
• For larger matrices, calculate the determinant using cofactor expansion or recursively reducing the matrix to smaller matrices
• Denote the determinant of matrix A as $det(A)$ or $|A|$
• The determinant of a 1x1 matrix is the value of its single element

Special Cases

• The determinant of a triangular matrix (upper or lower) is the product of the main diagonal elements
  • Example: For upper triangular matrix $A = \begin{bmatrix} a & b & c \ 0 & d & e \ 0 & 0 & f \end{bmatrix}$, $det(A) = adf$
• If a matrix has a row or column of zeros, its determinant is zero
• If a matrix has two identical rows or columns, its determinant is zero

Properties of Determinants

Transpose and Determinant

• The determinant of a matrix is equal to the determinant of its transpose: $det(A) = det(A^T)$
  • Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$ and $det(A) = det(A^T) = -2$

Row and Column Operations

• Interchanging two rows or two columns of a matrix changes the sign of the determinant
• Multiplying a row or column of a matrix by a scalar $k$ multiplies the determinant by $k$
  • Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 4 \ 3 & 4 \end{bmatrix}$ (first row of A multiplied by 2), then $det(B) = 2 \cdot det(A)$

Matrix Products and Invertibility

• The determinant of a product of matrices is equal to the product of their determinants: $det(AB) = det(A) \times det(B)$
• If a matrix is invertible (non-singular), its determinant is non-zero
  • Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $det(AB) = det(A) \times det(B) = -2 \times -2 = 4$

Cramer's Rule for Systems

Solving Linear Systems

• Cramer's Rule is a method for solving systems of linear equations using determinants
• For a system of $n$ linear equations with $n$ unknowns, Cramer's Rule states that the value of each variable can be found by dividing the determinant of a modified matrix by the determinant of the coefficient matrix
  • Example: For the system $\begin{cases} ax + by = e \ cx + dy = f \end{cases}$, $x = \frac{det(\begin{bmatrix} e & b \ f & d \end{bmatrix})}{det(\begin{bmatrix} a & b \ c & d \end{bmatrix})}$ and $y = \frac{det(\begin{bmatrix} a & e \ c & f \end{bmatrix})}{det(\begin{bmatrix} a & b \ c & d \end{bmatrix})}$
• To find the value of variable $x_i$, replace the $i$-th column of the coefficient matrix with the constant terms from the right side of the equations and calculate the determinant, then divide this determinant by the determinant of the original coefficient matrix

Applicability and Efficiency

• Cramer's Rule is applicable only when the system has a unique solution, which means the determinant of the coefficient matrix is non-zero
• While Cramer's Rule is straightforward, it becomes computationally inefficient for large systems of equations compared to other methods like Gaussian elimination
  • Example: Solving a 3x3 system using Cramer's Rule requires calculating four 3x3 determinants, while Gaussian elimination can solve the system with fewer operations

Geometric Interpretation of Determinants

Area and Volume

• The absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the row or column vectors of the matrix
  • Example: For matrix $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, $|det(A)| = 2$ represents the area of the parallelogram formed by vectors $(1, 3)$ and $(2, 4)$
• The absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the row or column vectors of the matrix
• In higher dimensions, the absolute value of the determinant represents the $n$-dimensional volume of the hyperparallelepiped formed by the row or column vectors

Orientation and Invertibility

• The sign of the determinant indicates the orientation of the linear transformation represented by the matrix
  • A positive determinant means the orientation is preserved
  • A negative determinant means the orientation is reversed
• If the determinant of a matrix is zero, the matrix is not invertible, and the linear transformation it represents maps the space to a lower-dimensional subspace, causing a collapse in dimension
  • Example: The matrix $A = \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix}$ has $det(A) = 0$, so it is not invertible and maps the 2D plane to a 1D line