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The reduced row-echelon forms of the augmented matrices of three systems are given below. How many solutions does each system have?

The reduced row-echelon forms of the augmented matrices of three systems are given in the attachment. How many solutions does each system have?

1. The reduced row-echelon forms of the augmented matrices of three systems are given below. How many solutions does each system have?

a. │1 0 2 0│
│0 1 3 0│
│0 0 0 1│

b. │1 0 5│
│0 1 6│

c. │0 1 0 2│
│0 0 1 3│

4. Find the rank of the matrix.

│ 1 4 7 │
│ 2 5 8 │
│ 3 6 9 │

36. Find a 3x3 matrix A such that

┌ ┐ ┌ ┐
│ 1 │ │ 1 │
A = │ 0 │ = │ 2 │,
│ 0 │ │ 3 │
└ ┘ └ ┘

┌ ┐ ┌ ┐
A = │ 0 │= │ 4 │ , and
│ 1 │ │ 5 │
│ 0 │ │ 6 │
└ ┘ └ ┘

┌ ┐ ┌ ┐
│ 0 │ │ 7 │
A = │ 0 │= │ 8 │
│ 1 │ │ 9 │
└ ┘ └ ┘

58. For which values of the constants b and c is the vector

┌ ┐
│3│ a linear combination of
│b│
│c│
└ ┘
┌ ┐
│1│
│3│,
│2│
└ ┘
┌ ┐
│2│
│6│, and
│4│
└ ┘
┌ ┐
│-1│
│-3│ ?
│-2│
└ ┘

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