Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 04/Solution 1

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We reduce the matrix to row echelon form. Subtracting the first row from the last row transforms

into

Next, subtracting times the second row from the third row yields

Clearly, this matrix has rank at least 2 (since, for instance, the first and third columns are linearly independent regardless of the value of ). On the other hand, it must have rank less than 3 whenever or (or both), which is to say that has rank 2 exactly when or (or both).