Question

In: Advanced Math

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the...

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the maximum number of linearly independent rows in A is 3. Discuss: Is the solution of the system unique?

Expert Solution

Since the system is consistent, there is at least one solution, say . Let be any solution to , and let ; then

Thus, .

Since the maximum number of linearly independent rows in is , we know that is the row-rank of . Since row-rank and column-rank of a matrix are identical, we conclude that column-rank of is also . Thus, all the columns of are linearly independent. Since the vector is a linear combination of the columns of , namely, , where and are columns of , we conclude that implies . This implies

Thus, if are solutions to then ; hence, the system has a unique solution.

Colby Messinger answered 2 years ago

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