CRAMER'S RULE, SARRUS' RULE, LAPLACE'S RULE AND INVERSE MATRIX

Fundamental tools in linear algebra to solve systems of equations and operate with matrices.
Cramer's rule
Solve systems of linear equations:

Ax = b

where:

1. *A*: Coefficient matrix
2. *x*: Vector of unknowns
3. *b*: Vector of constants

x_i = det(A_i) / det(A)

where:

1. *A_i*: Matrix obtained by replacing column i of A with b
Sarrus rule
Calculate the determinant of a 3x3 matrix:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

where:

1. *a, b, c, d, e, f, g, h, i*: Matrix elements
Laplace's rule
Calculate the determinant of an nxn matrix:

det(A) = ∑ (-1)^(i+j) a_ij M_ij

where:

1. *a_ij*: Element in row i, column j
2. *M_ij*: Lesser than a_ij
Inverse matrix
Matrix that multiplies another matrix to obtain the identity matrix:

A^-1 A = I

where:

1. *A^-1*: Inverse matrix of A
2. *I*: Identity matrix
Properties
1. (AB)^-1 = B^-1 A^-1
2. (A^-1)^-1 = A
3. det(A^-1) = 1 / det(A)
Applications
1. Resolution of systems of linear equations
2. Analysis of dynamic systems
3. Linear transformations
4. Vector algebra