Determine the value of h such that the matrix is the augmented matrix of a consistent linear system. I need help with both of the problems below.
1. [ 6 4 h] [-3 -2 7]
2. [1 2 -4] [3 h -12]
Answer
![Consider the augmented matrix, 1 2 A3h-12 RR-3R 1 2 -4 0 h-6 0 From the last matrix, observe that, rank(A)-rank([A b]) for an](https://fornoob.com/wp-content/uploads/2022/01/e1312e5bc8bf.png "e1312e5bc8bf")
Consider the augmented matrix, -3 -27 From the last matrix, observe that, rank(A)-1 and rank (A b) lif 14+h0 Recall the fact that, the system Ax-b is consistent f and only if rank (A) rank([A b]) In this problem, rank (4)-rank(lA | b)fH-h-o Hence, the linear system Ax bwith the given augmented matrix is consistent if h--14
Consider the augmented matrix, 1 2 A3h-12 RR-3R 1 2 -4 0 h-6 0 From the last matrix, observe that, rank(A)-rank([A b]) for any value of h Hence, the linbar system Ax=b with the given augmented matrix is consistent for all values of h
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