5:In each part below, give a m × n matrix R in reduce rou-echelon form satisfyin

Question

5:In each part below, give a m × n matrix R in reduce rou-echelon form satisfying the given condition, or explain briefly why it is impossible to do so. (a) m = 3, n 4, and the equation Rx = c has a solution for all c. (b) m = 3, n = 4, and the equation Rx = 0 has a unique solution. (c) m = 4, n = 3, and the equation Rx = c has a solution for all c. (d) m = 4, n = 3, and the equation Rx = 0 has a unique solution. (e) m = 4, n = 4, and the equation Rx-0 has no solution. (f) m-4, n = 4, and the equation Rx-0 has a nontrivial solution. (g) m 4, n = 4, and for every c the equations Rx-c have a solution containing a free paramete (w

Explanation / Answer

(a) m = 3 , n = 4.

Rx = c where R is in row echelon form.

Here order of R is 3×4.

Hence it is consistent in c. Hence has solution for all c.

(b) From rank nullity theorem --

Rank+nullity = 4

Rank of R is min(3,4) = 3

3+nullity = 4

Nullity = 1

Hence here nullity is only 1.so only unique solution is possible.

(c) and (d) has same argument because here only m and n have been interchanged only. It doesn't affect the rank and nullity.

(e) m = 4, n = 4

Here rank = min (4,4) = 4

From rank nullity theorem ---

Rank+nullity = 4

4+nullity = 4

Nullity = 0

Hence there is no solution of Rx = 0.

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