plz help me with this one and explain each step Suppose the system to the right

Question

plz help me with this one and explain each step

Suppose the system to the right is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer X1+ 12x2 f 2F 9 A. 12 0 d-12c g-cf and g, the second row could correspond to an equation of 0-b, where b is nonzero. Thus, d 1 12 f The row reduction of to shows that d - 12c# 0 since f and g are arbitrary. Otherwise, for some choices of f c d g 12c. O B. Since fand g are arbitrary, c and d can be any value and the system will be consistent Oc. 1 12 f 12 The row reduction of to shows that d-12c= 0 since f and g are arbitrary. Otherwise, for some choices of f c d g 0 d 12c g -cf and g, the second row could correspond to an equation of bx2_0 where b is nonzero. Thus, c=d =0 ? D. Since f and g are arbitrary and the system is consistent for all possible values off and g, c=0 and d = 1. Otherwise, the triangular form of the matrix will show that the system is inconsistent.

Explanation / Answer

If the given linear system is consistent for all the values of f and g, it implies that the columns of the given matrix A (say), span R2. Therefore, regardless of the values of c and d, the system is consistent. Hence the option B is the correct answer.

Note:

If we row reduce A as given in the question, and if the given linear system is consistent, then either d-12c and g-cf are both zero or both are non-zero. Otherwise, the given linear system will be inconsistent.

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