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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2 Answers
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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
- the leading columns of $A$ are also leading columns of $(\,A\mid b\,)$;
- so the two ranks are different if and only if the column $b$ becomes a leading column after reduction;
- ...if and only if there is a row of the form $(\,0\,\cdots\,0\mid c\,)$ with $c\ne0$;
- ...if and only if there is no solution.
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The theorem is obvious if the original matrix is in row-reduced echelon form. Reducing it to that form doesn't change the rank, and it doesn't change whether the system has solutions. So it's true for any matrix.
