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4.2 Linear Systems and Augmented Matrices
Question: What do these three systems of linear equations have in common? { x − 2 y = 2 x + y = 5
x + 2y = − 4 2 x + 4y = 8
2 x + 4y = 8 x + 2y = 4
Answer: They all have the same ”structure” and differ only in coef- ficients of x-s and y-s, and constants terms in the right-hand side.
Question: Given that any system of linear equations in two variables have the same structure, is there a way to represent a particular system in a more concise way?
Answer: We can use a mathematical form called a matrix:
Definition (Matrix) A matrix is a rectangular array of numbers written within brack- ets. Each number in a matrix is called an element of the matrix. If a matrix has m rows and n columns, it is called an m × n matrix (read “m by n matrix”). The expression m × n is called the size of the matrix, and the numbers m and n are called the dimensions of the matrix.
For example,
Definition (Square Matrix, Row Matrix, Column Ma- trix) A matrix with n rows and n columns is called a square matrix of order n. A matrix with only 1 column is called a column matrix, and a matrix with only 1 row is called a row matrix.
For example,
Solving Linear Systems Using Augmented Ma-
trices
Matrices serve as a shorthand for solving systems of linear equations by elimination.
Definition (Coefficient, Constant and Augmented Ma- trices) Given a linear system in two variables { ax + by = h cx + dy = k
the coefficient matrix of (1) is the matrix [ a b c d
]
the constant matrix of (1) is the column matrix [ h k
]
and the augmented matrix of (1) is the matrix [ a b h c d k
]
For example,
Recall that two linear systems are said to be equivalent if they have the same solution set. We can extend this definition to augmented matrices.
Definition (Row Equivalent Matrices) Two augmented matrices are said to be row equivalent, denoted by the symbol ∼ placed between the two matrices, if they are augmented matrices of equivalent systems of equations.
Naturally, we can reformulate Theorem 2 from Section 4.1 in terms of matrices.
Theorem 1 (Operations That Produce Row Equiva- lent Matrices)
An augmented matrix is transformed into a row equivalent matrix by performing any of the following row operations:
(a) two rows are interchanged (Ri ↔ Rj);
(b) a row is multiplied by a nonzero constant (kRi → Ri);
(c) a constant multiple of one row is added to another row; (kRj + Ri → Ri).
Example 2 Solve using augmented matrix methods: { − 2 x 1 + 6x 2 = 6 3 x 1 − 9 x 2 = − 9
Example 3 Solve using augmented matrix methods: { 2 x 1 − x 2 = 3 4 x 1 − 2 x 2 = − 1
