\displaystyle 1.\ \text{A system of }n\text{ simultaneous linear equations in }n\text{ unknowns }x_{1},x_{2},x_{3},\ldots ,x_{n}\text{ is}
\displaystyle a_{11}x_{1}+a_{12}x_{2}+\ldots +a_{1n}x_{n}=b_{1}
\displaystyle a_{21}x_{1}+a_{22}x_{2}+\ldots +a_{2n}x_{n}=b_{2}
\displaystyle \ldots
\displaystyle a_{n1}x_{1}+a_{n2}x_{2}+\ldots +a_{nn}x_{n}=b_{n}
\displaystyle \text{This system of equations can be written, in matrix form, as}
\displaystyle \begin{bmatrix} a_{11}&a_{12}&\ldots &a_{1n}\\ a_{21}&a_{22}&\ldots &a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\ldots &a_{nn} \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{bmatrix}=\begin{bmatrix} b_{1}\\ b_{2}\\ \vdots\\ b_{n} \end{bmatrix}
\displaystyle \text{or, }AX=B,\ \text{where }A=\begin{bmatrix} a_{11}&a_{12}&\ldots &a_{1n}\\ a_{21}&a_{22}&\ldots &a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\ldots &a_{nn} \end{bmatrix},\ X=\begin{bmatrix} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{bmatrix}\text{ and }B=\begin{bmatrix} b_{1}\\ b_{2}\\ \vdots\\ b_{n} \end{bmatrix}

\displaystyle 2.\ \text{A set of values of the variable }x_{1},x_{2},\ldots ,x_{n}\text{ satisfying all the equations simultaneously}
\displaystyle \text{is called a solution of the system.}

\displaystyle 3.\ \text{If a system of equations has one or more solutions, then it is said to be a consistent system}
\displaystyle \text{of equations, otherwise it is an inconsistent system of equations.}

\displaystyle 4.\ \text{A system of equations }AX=B\text{ is called a homogeneous system, if }B=O.\text{ Otherwise, it is} \\ \text{called a non-homogeneous system of equations.}

\displaystyle 5.\ \text{A system }AX=B\text{ of }n\text{ linear equations in }n\text{ unknowns has a unique solution given by}
\displaystyle X=A^{-1}B,\ \text{if }|A|\neq 0.
\displaystyle \text{If }|A|=0\text{ and }(\mathrm{adj}\ A)B=O,\text{ then the system is consistent and has infinitely many solutions.}
\displaystyle \text{If }|A|=0\text{ and }(\mathrm{adj}\ A)B\neq O,\text{ then the system is inconsistent.}

\displaystyle 6.\ \text{A homogeneous system of }n\text{ linear equations in }n\text{ unknowns is expressible in the form}
\displaystyle AX=O.
\displaystyle \text{If }|A|\neq 0,\text{ then }AX=O\text{ has unique solution }X=O\text{ i.e. }x_{1}=x_{2}=\ldots =x_{n}=0. \\ \text{This solution is}  \text{called the trivial solution.}
\displaystyle \text{If }|A|=0,\text{ then }AX=O\text{ has infinitely many solutions.}

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