• Let $a_ij, b_j \in \mathbb{R}$ for i = 1, 2, 3, ….,m and j = 1, 2, 3, …, n. Afinite set of linear equations:

$\begin{cases} a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + … + a_{1n}x_{n} = b_{1} \\
a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} + … + a_{2n}x_{n} = b_{2}\\a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} + … + a_{3n}x_{n} = b_{3}\\ …\\…\\a_{m1}x_{1} + a_{m2}x_{2} + a_{m3}x_{3} + … + a_{mn}x_{n} = b_{m}\end{cases}$ is said to be a system of linear equations with n variables $x_{1}, x_{2}, x_{3},…., x_{n}$

• The above system of linear equation can also be written as:

AX = B, where $A = \begin{pmatrix}a_{11}&a_{12} &a_{13} & … a_{1n}\\a_{21} &a_{22}&a-{31}& ….a_{2n}\\.\\.\\.\\a_{m1}& a_{m2}&a_{m3}….a_{mn}\end{pmatrix}$ is the coefficient matrix

$X = \begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\.\\.\\x_{n}\end{pmatrix}$ is the variable vector whereas $B =\begin{pmatrix}b_{1} \\b_{2}\\b_{3}\\.\\.\\.\\b_{m}\end{pmatrix}$ is the constant vector(matrix)

• Gauss used elementary row / column operations on augmented matrix
• Swipping :- interchanging rows of a matrix$R_{i}\leftrightarrow R_{j}$
• Re-scaling: – multiply a row of a matrix by non zero constant $R_{i}\rightarrow kR_{i}$
• Pivoting: – add constant multiple of one row to an other row $R_{j}\rightarrow R_{j} +kR_{i}$

Unique (one ) solution: A linear system has unique solution if and only if it has only one solution.

Many(infinite) solution: The last row of reduced augmented matrix is zero. Graph of equations coincide to each other

No Solution ( Inconsistent): The system has no set of values satisfying the equations symaultinously. The lines do not intersect  at all.