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# System of Linear Equations in Matrix Form

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:

a11x1 + a12x2 + a13x3 + ŌĆ”ŌĆ”ŌĆ” + a1nxn + = b1

a21x1 + a22x2 + a23x3 + ŌĆ”ŌĆ”ŌĆ” + a2nxn + = b2

a31x1 + a32x2 + a3x3 + ŌĆ”ŌĆ”ŌĆ” + a3nxn + = b3

ŌĆ”..    ŌĆ”.   ŌĆ”..   ŌĆ”..   ŌĆ”..    ...

Am1x1 + am2x2 + am3x3 + ŌĆ”ŌĆ”ŌĆ” + amnxn + = bm

where the coefficients  aij , i = 1, 2, ŌĆ”. , m; j = 1, 2,ŌĆ”.., n  and bk , k = 1, 2,ŌĆ”.., m   are constants. If all  the bk 's are zeros, then the above system is called a homogeneous system of linear equations. On the other hand, if at least one of the bk 's is non-zero, then the above system is called a non-homogeneous system of linear equations. If there exist values ╬▒1 , ╬▒2 , ŌĆ”.. , ╬▒n for x1, x2 , ŌĆ”. , xn respectively which satisfy every  equation of (1), then the ordered  n ŌłÆ tuple (╬▒1 ,  ╬▒2 , ŌĆ”. , ╬▒n ) is called a solution of (1). The above system (1) can be put in a matrix form as follows:

Let A = be   the m x n matrix formed by the coefficients of x1, x2 , x3,ŌĆ”. , xn . The first row of A is formed by the coefficients of x1, x2 , x3,ŌĆ”. , xn 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 x1 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 x1, x2 , x3,ŌĆ”. , xn

Let   B  = be the m x 1 order column matrix formed by the right-hand side constants b1, b2 , b3 , ŌĆ”. , bm .

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