Answer to Question #176506 in Linear Algebra for Azuzile

Question

Consider the following system of linear equations:

x − y + z = 1

x + y − 2z = 2

2x − z = 3

(a) How many solutions does the system have? Justify your

answer

Answer 1

Let us find the rank of the extended matrix of the system of equations:

$\left( {\left. {\begin{matrix} 1&{ - 1}&1\\ 1&1&{ - 2}\\ 2&0&{ - 1} \end{matrix}} \right|\begin{matrix} 1\\ 2\\ 3 \end{matrix}} \right)\begin{matrix} {}\\ { - I}\\ { - 2I} \end{matrix} \sim \left( {\left. {\begin{matrix} 1&{ - 1}&1\\ 0&2&{ - 3}\\ 0&2&{ - 3} \end{matrix}} \right|\begin{matrix} 1\\ 1\\ 1 \end{matrix}} \right)\begin{matrix} {}\\ { - II} \end{matrix} \sim \left( {\left. {\begin{matrix} 1&{ - 1}&1\\ 0&2&{ - 3}\\ 0&0&0 \end{matrix}} \right|\begin{matrix} 1\\ 1\\ 0 \end{matrix}} \right)$

Since the ranks of the main and extended matrix are equal to each other, but not equal to the number of equations:

$rank(A) = rank(A|B) = 2 \ne 3$ , the system of equations is consistent, but not defined, and therefore has an infinite number of solutions