What is an augmented matrix
An augmented matrix is a way to represent a system of linear equations using a matrix
An augmented matrix is a way to represent a system of linear equations using a matrix. It includes both the coefficient matrix and the constant vector.
In a system of linear equations, each equation can be represented as:
a₁₁x₁ + a₁₂x₂ + … + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + … + a₂ₙxₙ = b₂
…
aₘ₁x₁ + aₘ₂x₂ + … + aₘₙxₙ = bₘ
Here, the coefficients a₁₁, a₁₂, …, aₘₙ represent the constants multiplying the variables, and the constants b₁, b₂, …, bₘ are the right-hand side of the equations.
To form the augmented matrix, the coefficients a₁₁, a₁₂, …, aₘₙ and the constants b₁, b₂, …, bₘ are combined into a single matrix. The augmented matrix has m rows and (n + 1) columns.
It looks like this:
[ a₁₁ a₁₂ … a₁ₙ | b₁ ]
[ a₂₁ a₂₂ … a₂ₙ | b₂ ]
[ … ]
[ aₘ₁ aₘ₂ … aₘₙ | bₘ ]
The augmented matrix allows us to perform row operations, such as row addition, row multiplication, and row swapping, to manipulate and solve the system of linear equations efficiently.
More Answers:
Understanding Existence and Uniqueness in Linear Systems of Equations | Essential Concepts for Solving and AnalyzingUnderstanding Row Equivalence in Linear Algebra | A Guide to Matrices and Elementary Row Operations
Mastering Row Equivalence in Matrices | Transforming and Simplifying with Elementary Row Operations