Coefficient matrix

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A coefficient matrix is the matrix that holds all the coefficients of the variables in a system of linear equations. It helps us solve the system.

Suppose there are m equations and n unknowns. The unknowns are x1, x2, ..., xn, and the numbers a11, a12, ..., amn are the coefficients. The coefficient matrix A is the m×n matrix with entry aij in row i, column j. The constants on the right-hand side form a column vector b.

The system can be written concisely as A x = b, where x is the column vector of unknowns.

Solvability (Rouché–Capelli theorem):
- The system is inconsistent (no solution) if the rank of the augmented matrix [A | b] is greater than the rank of A.
- If the ranks are equal, the system has at least one solution.
- The solution is unique exactly when the rank r of A equals the number of unknowns n.
- If r < n, there are infinitely many solutions. There are n − r free parameters; you can fix some variables and solve for the rest, with different choices giving different solutions.

First-order matrix difference equation (constant term):
- It can be written as x_{t+1} = A x_t + c, where A is n×n and c is an n×1 vector.
- This system converges to a steady state if every eigenvalue of A has absolute value less than 1.

First-order matrix differential equation (constant term):
- It can be written as dy/dt = A y + c.
- This system is stable if every eigenvalue of A has a negative real part.