Describe what is meant by the augmented matrix of a system of linear equations.

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Answer 1

Step 1
Describe what is meant by the augmented matrix of a system of linear equations.
Step 2
Let us consider a linear system of N equations in M unknowns

A_{11} x_{1} + A_{12} x_{2} + A_{13} x_{3} + \dots \dots . . + A_{1 M} x_{M} = b_{1}

A_{21} x_{1} + A_{22} x_{2} + A_{23} x_{3} + \dots \dots . . + A_{2 M} x_{M} = b_{2}

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A_{N 1} x_{1} + A_{N 2} x_{2} + A_{N 3} x_{3} + \dots \dots . . + A_{N M} x_{M} = b_{N}

represnted as , Ax=b where A is the

N \times M

matrix of coefficients,x is the

M \times 1

vectors of unknowns and b is the

N \times 1

vector of constantThe augmented matrix of the system is

(A ∣ b)

e.g, The system of two equations in two unknowns

2 x_{1} + x_{2} = 1

x_{1} + 2 x_{2} = 3

can be represented in matrix form as
Ax=b
where

A = [\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}], x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] and b = [\begin{matrix} 1 \\ 3 \end{matrix}]

The augmented matrix of the system is

\Rightarrow (A ∣ b) = [\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 3 \end{array}]

Describe what is meant by the augmented matrix of a system of linear equations.

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