A 3x3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions. These examples illustrate a theorem about linear combinations of the columns of the matrix A.

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##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z

##### Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.row_reduce.i

##### The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb

##### Definition of matrix equation math.la.d.mat.eqn