Problem 4 - Algebra 2

Question

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{aligned} x+3 y-z &=2 \\ x &+2 z=8 \\ 2 y-z &=1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The coefficient matrix is \(A = \begin{bmatrix} 1 & 3 & -1 \ 1 & 0 & 2 \ 0 & 2 & -1 \ \end{bmatrix}\), the variable matrix is \(\mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}\), and the constant matrix is \(\mathbf{b} = \begin{bmatrix} 2 \ 8 \ 1 \end{bmatrix}\). Therefore, the matrix equation is \(A \mathbf{x} = \mathbf{b}\).

Step 1: Writing the system as a matrix equation

First, we need to identify the coefficients of the variables in each equation of the system and use those to create the coefficient matrix. Next, create a column matrix for the variables, and another for the constants on the right-hand side.

Step 2: Identifying the coefficient matrix

The coefficient matrix is composed of the coefficients of the variables x, y, and z in each equation, placed in the same order. In our case this will be a 3x3 matrix since we have three equations with three variables.

Step 3: Identifying the variable matrix

The variable matrix (also known as the vector of unknowns) is a column matrix that contains the variables of the system. Each variable is placed in a separate row.

Step 4: Identifying the constant matrix

The constant matrix is a column matrix formed by the constants from the right-hand side of the equations of the system.

Step 5: Assembling the matrix equation

Combine the coefficient matrix, the variable matrix, and the constant matrix to form the matrix equation \(A \mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the variable matrix, and \(\mathbf{b}\) is the constant matrix.

Key Concepts

Coefficient Matrix

To start with, the coefficient matrix is a crucial element when dealing with systems of linear equations. It brings organization and structure to the system by capturing all the coefficients in a compact, easy-to-manipulate form. Imagine lining up soldiers: each coefficient is a soldier and the coefficient matrix is the formation they stand in. In the provided exercise, we can see different 'soldiers' aligned in distinct patterns across the equations. Each equation in the system contributes a row to the matrix, and each variable corresponds to a column.
When we list these coefficients in a matrix, we get something looking like this:
\[A = \begin{bmatrix}1 & 3 & -1 \1 & 0 & 2 \0 & 2 & -1\end{bmatrix}\]
This matrix, labelled as \(A\), is essential because it captures the essence of the system's structure without involving the variables or the solutions, making it easier to perform mathematical operations and solve for the unknown variables.

Variable Matrix

When it comes to the variable matrix, think of it as a container that holds all unknown factors you are trying to find. In our exercise where we're involved with a system of equations, the variable matrix is akin to a shopping list, outlining exactly what items (in this case, variables) we're on the lookout for.
For the given system of equations, the variable matrix is crafted as follows:
\[\mathbf{x} = \begin{bmatrix}x \y \z\end{bmatrix}\]
This vertical array, with one column and as many rows as there are variables, reveals the order in which we're considering our unknowns. In computations, this matrix will be multiplied by the coefficient matrix to contribute to the quest for the solution.

Constant Matrix

The constant matrix is like the bus stop where all the numerical values that don't have variables attached are waiting. It's where, after aligning all variables and coefficients in a neat row and column array, the constants stand on the right side of the equation, anticipating their matching variables to be paired with. In many ways, these constants tell a story about the resources available in our problem.
In our specific example, the constants from the equations compile to create the following matrix:
\[\mathbf{b} = \begin{bmatrix}2 \8 \1\end{bmatrix}\]
This matrix is always a column matrix, directly corresponding to the shape of the variable matrix. It is separate from the coefficient matrix but plays a symbiotic role as we move forward to solve the matrix equation.

System of Linear Equations

When we talk about a system of linear equations, we're referring to a collection of equations that we aim to solve simultaneously. It's somewhat like a puzzle where each piece must fit with the others to see the whole picture—at least one set of values, if they exist, that satisfies all equations simultaneously.
Our original system can now be represented in a compact and mathematical form as a matrix equation by combining the coefficient matrix, the variable matrix, and the constant matrix:
\[A\mathbf{x} = \mathbf{b}\]
Here's the big reveal: using the matrices from our exercise, our matrix equation looks like this:
\[\begin{bmatrix}1 & 3 & -1 \1 & 0 & 2 \0 & 2 & -1\end{bmatrix} \begin{bmatrix}x \y \z\end{bmatrix} = \begin{bmatrix}2 \8 \1\end{bmatrix}\]
What we've done is distill the entire system into a single, elegant equation. This nifty transformation not only streamlines our work but also unlocks powerful methods for finding solutions, such as using matrix operations or inversions, all within reach thanks to this algebraic shorthand.