Are there values of r and s for which the rank of [1 0 0 0 r - 2 2 0 s - 1 r + 2

Question

Are there values of r and s for which the rank of [1 0 0 0 r - 2 2 0 s - 1 r + 2 0 0 3] is one? two? If so, find those values. Determine whether each statement is true or false. If a statement is true, then give a reason or cite an appropriate statement in the text. If a statement is false, then provide a counterexample. (a) If A is not square, then the row vectors of A must be linearly dependent. (b) If A is square, then either the row vectors or the column vectors of A must be linearly independent. (c) If the row vectors and the column vectors of A are linearly independent, then A must be square. (d) Adding one additional column to a matrix A always increases its rank by one.

Explanation / Answer

4) there is no such value of r and s which will make the rank of the matrix 1. As there is 1 in the (1,1) position and 3 at (4,3) position.

If we take s=1 and r=2 then the second column becomes zero . And thus there will be two nonzero columns .

Therefore rank is 2. In the case of s=1,r=2

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