What is a linear system?

A set of equations in matrix form Ax=b where A is coefficient matrix, x is unknown vector, b is constants.

If A is invertible: x = A^-1 * b. Or use Cramer rule: x_i = det(A_i)/det(A).

When is there no solution?

When det(A)=0 (singular matrix). System has either no solutions or infinite solutions.

What is Cramers rule?

Replace column i of A with b, compute determinant, divide by det(A) to get x_i.

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Linear System Solver

Solve Ax = b for 2x2 and 3x3 systems of equations

Matrix A (coefficients)

Vector b (constants)

System Solving Method

Ax = b → x = A^(-1) * b (if A invertible)

Cramer rule: x_i = det(A_i) / det(A)

Check: A * x should equal b

No solution if det(A) = 0

A linear system has a unique solution when the coefficient matrix is invertible. If det=0, check consistency.

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