(1 point) Let A be a matrix with linearly independent columns Select the best st

Question

(1 point) Let A be a matrix with linearly independent columns Select the best statement A. The equation Ax b has a solution for all b precisely when it has more rows than columns. B. The equation Ax b has a solution for all b precisely when it has more columns than rows C. The equation Ax = b has a solution for all b precisely when it is a square matrix. D. The equation A - b always has a solution for all b E. The equation Ax- b never has a solution for all b. F. There is no easy way to tell if Ax -b has a solution for all b. G. none of the above

Explanation / Answer

Q.

Since Ax=b is actually a way to solve linear equations in n variables. And it has solution,

when the row echeleon form of the augmented matrix does not have a leading 1 in the column that corresponds to constants.

In this case, Since the columns of the matrix are linearly independent then it means that the number of columns is not more than the number of rows, which means option (2) is not possible.

Now, if the matrix has more rows than columns then it can be possible to have a b for which there is no solution so option (1) is not always true.

Hence, the only true option is option (c), when A will be a square matrix.

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