Rank of a Matrix and Existence of a Unique Solution to an Equations System
             The rank of a matrix is the maximum number of linearly independent rows or columns in it. Alternatively, it is the order of the largest square sub-matrix possible with a non-zero determinant. If the matrix is of order m‘n, then the rank cannot be greater than the smaller of m or n. Hence, to reduce search time for linear independence, choose to investigate either the set of columns or the set of rows depending on whichever dimension is smaller. For a square matrix, choice doesn't matter for search time.
             Consider the square matrix, A = whose determinant0, i.e. rank = 3 as all three columns or rows are linearly independent. Now consider the equations system, = . This system has the solution: x = -33, y = -3 and z = 19, i.e. = , i.e. , i.e. is a linear combination of the columns of A with the solutions as the weights. Suppose that you formed an augmented matrix B by taking the three columns of A and as the fourth column, i.e. B = . What would be the rank of B, i.e. the maximum number of linearly independent columns in B? We hope you answered 3.
             Can you now see that an equations system Ax = b has a solution only if rank(A) = rank(A,b)?
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