Applications of Matrices: Solving System of Linear Equations

**System of Linear Equations in Matrix Form**

A system of *m *linear equations in *n *unknowns is of
the following form:

*a*_{11}*x*_{1} +* a*_{12}*x*_{2} +* a*_{13}*x*_{3}
+ â€¦â€¦â€¦ +* a*_{1n}*x*_{n} + = *b*_{1}

*a*_{21}*x*_{1} +* a*_{22}*x*_{2} +* a*_{23}*x*_{3}
+ â€¦â€¦â€¦ +* a*_{2n}*x*_{n} + = *b*_{2}

*a*_{31}*x*_{1} +* a*_{32}*x*_{2} +* a*_{3}*x*_{3}
+ â€¦â€¦â€¦ +* a*_{3n}*x*_{n} + = *b*_{3}

â€¦.. â€¦. â€¦.. â€¦.. â€¦.. ...

*A*_{m1}*x*_{1} +* a*_{m2}*x*_{2} +* a*_{m3}*x*_{3}
+ â€¦â€¦â€¦ +* a*_{mn}*x*_{n} + = *b*_{m}

where the coefficients a_{ij} , i = 1, 2, â€¦. , *m*; j = 1, 2,â€¦.., *n* and *b _{k} *, k = 1, 2,â€¦..,

Let A = be
the *m *x* n* matrix formed by the coefficients of x_{1}, *x*_{2} , *x*_{3},â€¦. , *x*_{n}
. The first row of A is formed by the coefficients of x_{1}, *x*_{2} , *x*_{3},â€¦. , *x*_{n}
in the same order in which they occur in the first equation. Likewise, the
other rows of *A* are formed. The first
column is formed by the coefficients of *x*_{1}
in the *m* equations in the same order.
The other columns are formed in a similar way.

Let X = be
the *n* x1 order column matrix formed
by the unknowns x_{1}, *x*_{2}
, *x*_{3},â€¦. , *x*_{n}

Let B = be the m x 1 order column matrix
formed by the right-hand side constants *b*_{1},
*b*_{2} , *b*_{3} , â€¦. , *b*_{m}
.

Then we get

Then AX = B is a matrix equation involving matrices and it is
called the **matrix form** of the system of
linear equations (1). The matrix A is called the **coefficient matrix** of the system and the
matrix

is called the **augmented matrix **of the system. We
denote the augmented matrix by [ A | B ].

As an example, the matrix form of the system of linear equations

2x + 3y - 5z + 7 = 0, 7 y + 2z - 3x = 17, 6x - 3y - 8z + 24 = 0
is

Tags : Applications of Matrices: Solving System of Linear Equations , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : System of Linear Equations in Matrix Form | Applications of Matrices: Solving System of Linear Equations

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