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

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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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