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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. &#9474;1 0 2 0&#9474;
&#9474;0 1 3 0&#9474;
&#9474;0 0 0 1&#9474;

b. &#9474;1 0 5&#9474;
&#9474;0 1 6&#9474;

c. &#9474;0 1 0 2&#9474;
&#9474;0 0 1 3&#9474;

4. Find the rank of the matrix.

&#9474; 1 4 7 &#9474;
&#9474; 2 5 8 &#9474;
&#9474; 3 6 9 &#9474;

36. Find a 3x3 matrix A such that

&#9484; &#9488; &#9484; &#9488;
&#9474; 1 &#9474; &#9474; 1 &#9474;
A = &#9474; 0 &#9474; = &#9474; 2 &#9474;,
&#9474; 0 &#9474; &#9474; 3 &#9474;
&#9492; &#9496; &#9492; &#9496;

&#9484; &#9488; &#9484; &#9488;
A = &#9474; 0 &#9474;= &#9474; 4 &#9474; , and
&#9474; 1 &#9474; &#9474; 5 &#9474;
&#9474; 0 &#9474; &#9474; 6 &#9474;
&#9492; &#9496; &#9492; &#9496;

&#9484; &#9488; &#9484; &#9488;
&#9474; 0 &#9474; &#9474; 7 &#9474;
A = &#9474; 0 &#9474;= &#9474; 8 &#9474;
&#9474; 1 &#9474; &#9474; 9 &#9474;
&#9492; &#9496; &#9492; &#9496;

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

&#9484; &#9488;
&#9474;3&#9474; a linear combination of
&#9474;b&#9474;
&#9474;c&#9474;
&#9492; &#9496;
&#9484; &#9488;
&#9474;1&#9474;
&#9474;3&#9474;,
&#9474;2&#9474;
&#9492; &#9496;
&#9484; &#9488;
&#9474;2&#9474;
&#9474;6&#9474;, and
&#9474;4&#9474;
&#9492; &#9496;
&#9484; &#9488;
&#9474;-1&#9474;
&#9474;-3&#9474; ?
&#9474;-2&#9474;
&#9492; &#9496;

#### Solution Summary

This solution is comprised of a detailed explanation to answer how many solutions does each system have.

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