### Systems of linear equations and matrices: Systems and matrices

### Solvability of systems of linear equations

Using the concept of *rank* for a matrix, we can characterize the solvability of a system of linear equations. First a definition of the key concept:

Rank The **rank** of a matrix is the number of non-zero rows in an *echelon form* of the matrix. We denote the rank of a matrix \(A\) by \(\text{rank}(A)\).

- A system of linear equations is inconsistent if and only if the rank of the coefficient matrix is smaller than the rank of the augmented matrix.
- If a system of #m# linear equations in #n# unknowns has a solution, then the solution is parameterized by \(n-r\) free parameters, where \(r\) is the rank of the associated coefficient matrix.
- If the coefficient matrix of a system of #m# linear equations in #n# unknowns has rank #n#, then the system has a unique solution.

For homogeneous systems we already know that there is always a solution:

Non-trivial solutions of homogeneous systems Each homogeneous system has a **trivial** solution in which all the values of the unknowns are equal to zero. A non-zero solution of a homogeneous system is called a **non-trivial** solution.

Each homogeneous system of linear equations having more unknowns than equations, possesses non-trivial solutions.

#\text{rank}(A)=# #2#

With the aid of elementary operations on the rows we reduce the matrix to reduced echelon form: \[ \begin{array}{rcl}A = \matrix{1&1&4&-4\\1&2&5&-3\\2&3&9&-7\\}&\sim\matrix{1&1&4&-4\\0&1&1&1\\2&3&9&-7\\}&{\color{blue}{\begin{array}{c}\phantom{x} R_2-R_1\phantom{x}\end{array}}}\\&\sim\matrix{1&1&4&-4\\0&1&1&1\\0&1&1&1\\}&{\color{blue}{\begin{array}{c}\phantom{x} R_3-2R_1\end{array}}}\\&\sim\matrix{1&0&3&-5\\0&1&1&1\\0&1&1&1\\}&{\color{blue}{\begin{array}{c}R_1-R_2\phantom{x}\end{array}}}\\&\sim\matrix{1&0&3&-5\\0&1&1&1\\0&0&0&0\\}&{\color{blue}{\begin{array}{c}\phantom{x} R_3-R_2\end{array}}}\end{array}\]

Because the rank is the number of non-null rows of this matrix, we conclude that the rank of the matrix #A# equals #2#.

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