What does the matrix A represent in the equation Ax = b?

In the equation ( Ax = b ), the matrix ( A ) is known as the coefficient matrix. This matrix is composed of the coefficients of the variables in the system of linear equations represented by the equation. In other words, each entry in matrix ( A ) corresponds to a specific coefficient that multiplies a corresponding variable in the vector ( x ).

The importance of the coefficient matrix lies in its role in transforming the variable vector ( x ) into the resultant vector ( b ) through matrix multiplication. Thus, when we multiply ( A ) by ( x ), we are essentially applying a linear transformation defined by ( A ) to the vector ( x ), which results in the vector ( b ).

Understanding that ( A ) represents the coefficients helps in grasping the structure of systems of linear equations and provides insights into solving them, whether through methods like Gaussian elimination, matrix inversion, or other algebraic techniques.