Final answer:
The given system of equations can be written as a matrix equation, a vector equation, and an augmented matrix. Each representation formats the information differently, but they are mathematically equivalent.
Step-by-step explanation:
We have been given a system of equations:
-
- x₁ − 3x₂ = −12
-
- 2x₁ + 2x₂ = −4
-
- −4x₁ + 5x₂ = 8
Let's proceed with the tasks:
a. Write this system as a matrix equation.
A matrix equation takes the form Ax = b, where A is the coefficient matrix, x is the column of variables, and b is the column of solutions. In this case:
A =
[
[1, -3],
[2, 2],
[-4, 5]
]
x =
[
[x₁],
[x₂]
]
b =
[
[-12],
[-4],
[8]
]
b. Write this system as a vector equation.
In vector format, each equation can be expressed as a linear combination of vectors. Here, we have:
x₁ * [
[1],
[2],
[-4]
] + x₂ * [
[-3],
[2],
[5]
] = [
[-12],
[-4],
[8]
]
c. Write this system as an augmented matrix.
The augmented matrix combines the coefficient matrix with the solutions matrix:
[1 -3 | -12]
[2 2 | -4]
[-4 5 | 8]