Can someone please explain why this theorem is true?
Theorem: If A is the matrix of coefficients of a system of linear equations, then the system has a solution if and only if the rank of the augmented matrix is equal to the rank of the matrix A?
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Consider the matrix $A$ and the augmented matrix $(\,A\mid b\,)$. Reduce to row-echelon form. The rank (for both of these) is the number of leading (pivot) columns. Now