Which System Of Equations Represents The Matrix Shown Below

Understanding which system of equations represents the matrix shown below is a fundamental skill in linear algebra that bridges the abstract world of matrices with the concrete language of equations. When you look at an augmented matrix, each row encodes a linear equation, and the columns (except the last one) correspond to the coefficients of the variables. By translating the matrix back into algebraic form, you can solve the system using substitution, elimination, or matrix methods, and you gain insight into the geometric interpretation of solutions—whether they intersect at a single point, lie along a line, or have no common solution at all. This article walks you through the concepts, procedures, and nuances needed to confidently answer the question “which system of equations represents the matrix shown below?” for any matrix you encounter.

1. What Is an Augmented Matrix?

An augmented matrix combines the coefficient matrix of a linear system with the constants from the right‑hand side of the equations, separated by a vertical bar or a dotted line. For a system with n variables, the matrix has n columns for coefficients and one extra column for the constants.

For example, the system

[ \begin{cases} 2x + 3y - z = 5\ 4x - y + 2z = 6\ -x + 2y + 3z = -4 \end{cases} ]

is represented by the augmented matrix

[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5\ 4 & -1 & 2 & 6\ -1 & 2 & 3 & -4 \end{array}\right]. ]

The left block (the first three columns) is the coefficient matrix; the rightmost column holds the constants. Recognizing this structure is the first step in answering “which system of equations represents the matrix shown below?”

2. Translating Rows into Equations

Each row of the augmented matrix corresponds to one linear equation. The entries in a row are the coefficients of the variables, in the same order as the variable list (usually (x_1, x_2, \dots, x_n)), followed by the constant term.

Step‑by‑step procedure

1. Identify the variable order – If the problem does not specify, assume the standard order (x, y, z, \dots) or (x_1, x_2, x_3, \dots).
2. Read a row – Take the first (n) numbers as coefficients, the last number as the constant.
3. Write the equation – Multiply each coefficient by its variable, add the terms, and set the sum equal to the constant.
4. Repeat for every row – You will obtain as many equations as there are rows.

If a coefficient is zero, the corresponding variable simply does not appear in that equation. If the constant column is missing (i.e., you have a plain coefficient matrix), the system is homogeneous and all constants equal zero.

3. Worked Examples

Example 1: A 2 × 2 Augmented Matrix

Matrix:

[ \left[\begin{array}{cc|c} 1 & 4 & 7\ -2 & 3 & -1\end{array}\right]. ]

• Row 1: coefficients (1) (for (x)) and (4) (for (y)), constant (7) → (1x + 4y = 7) or simply (x + 4y = 7).
• Row 2: coefficients (-2) (for (x)) and (3) (for (y)), constant (-1) → (-2x + 3y = -1).

Thus the system is

[ \begin{cases} x + 4y = 7\ -2x + 3y = -1 \end{cases}. ]

Example 2: A 3 × 3 Matrix with a Zero RowMatrix:

[ \left[\begin{array}{ccc|c} 2 & 0 & -5 & 10\ 0 & 3 & 6 & 0\ 0 & 0 & 0 & 0 \end{array}\right]. ]

• Row 1: (2x + 0y -5z = 10) → (2x - 5z = 10).
• Row 2: (0x + 3y + 6z = 0) → (3y + 6z = 0) → divide by 3: (y + 2z = 0).
• Row 3: All zeros → (0 = 0), which is always true and adds no information.

The system reduces to

[\begin{cases} 2x - 5z = 10\ y + 2z = 0 \end{cases}, ]

showing that (z) is a free variable and the solution set is a line in (\mathbb{R}^3).

Example 3: Inconsistent SystemMatrix:

[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 4\ 0 & 0 & 0 & 5\ 2 & -1 & 0 & 1 \end{array}\right]. ]

• Row 1: (x + 2y + 3z = 4).
• Row 2: (0x + 0y + 0z = 5) → (0 = 5), a contradiction.
• Row 3: (2x - y + 0z = 1) → (2x - y = 1).

Because the second row yields an impossible statement, the system has no solution; it is inconsistent.

4. Special Forms: Row‑Echelon and Reduced Row‑Echelon

When a matrix is in row‑echelon form (REF) or reduced row‑echelon form (RREF), the translation process becomes even more straightforward because leading coefficients (pivots) are isolated.

• In REF, each leading entry is to the right of the one above it, and rows of all zeros sit at the bottom.
• In RREF, each leading entry is 1 and is the only non‑zero entry in its column.

Example (RREF):

[ \left[\begin{array}{ccc|c} 1 & 0 & 2 & 5\ 0 & 1 & -1 & 3\ 0 & 0 & 0 & 0 \end{array}\right]. ]

Translates to

[ \begin{cases} x + 2z = 5\ y - z = 3 \end{cases