15 Aug, 2024

· Mathematics

Which system of equations is represented by the matrix below

\left(\begin{array}{ccc|c} 2 & -1 & -3 & -20 \\ -2 & -6 & 6 & -4 \\ 0 & 3 & -2 & -9 \end{array}\right)

Short Answer

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Long Explanation

Explanation

System of Equations Representation

The matrix given is:

\left(\begin{array}{ccc|c} 2 & -1 & -3 & -20 \\ -2 & -6 & 6 & -4 \\ 0 & 3 & -2 & -9 \end{array}\right)

This matrix corresponds to a system of linear equations. Each row in the matrix represents a single linear equation, with the coefficients of the variables on the left and the constant term on the right.

Given the matrix:

\left(\begin{array}{ccc|c} 2 & -1 & -3 & -20 \\ -2 & -6 & 6 & -4 \\ 0 & 3 & -2 & -9 \end{array}\right)

we can interpret it as a system of equations:

First Equation:

2x - y - 3z = -20

Second Equation:

-2x - 6y + 6z = -4

Third Equation:

3y - 2z = -9

Thus, the system of equations represented by the matrix is:

\begin{cases} 2x - y - 3z = -20 \\ -2x - 6y + 6z = -4 \\ 3y - 2z = -9 \end{cases}

Verified By

Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

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Concept

Matrix Representation Of Linear Systems

Explanation of Matrix Representation of Linear Systems

The matrix representation of linear systems is a powerful way to express and solve systems of linear equations using matrices and vectors. It is often more efficient and organized than dealing with each equation individually.

Representing Linear Systems in Matrix Form

A linear system typically consists of multiple linear equations. Here's a simple example with two equations:

\begin{align*} 2x + 3y &= 5 \\ 4x - y &= 3 \end{align*}

This system can be represented in matrix form as:

\mathbf{A}\mathbf{x} = \mathbf{b}

Where:

$\mathbf{A}$ is the coefficient matrix,
$\mathbf{x}$ is the variable vector,
$\mathbf{b}$ is the constant vector.

For the given example, we have:

\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 4 & -1 \\ \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 3 \\ \end{pmatrix}

In general, for a linear system with $m$ equations and $n$ variables, the matrix representation looks like:

\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}

Solving the System Using Matrix Methods

Once in matrix form, various methods can be applied to solve the system:

Gaussian Elimination: A series of row operations to transform the matrix into its row echelon form.
Matrix Inversion: If the matrix $\mathbf{A}$ is square and invertible, the solution can be found using $\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$ .
LU Decomposition: Decomposing $\mathbf{A}$ into lower and upper triangular matrices, making it easier to solve the system.
Iterative Methods: Useful for larger systems where direct methods are inefficient, such as the Jacobi or Gauss-Seidel methods.

Matrix representation provides a compact and structured way to handle complex linear systems, leveraging linear algebra techniques that simplify both the theoretical analysis and computational solutions.

Concept

Row Interpretation Of Matrices

Explanation

Matrix interpretation can be approached from different perspectives, one of which is row interpretation. Understanding the row interpretation helps in visualizing and solving systems of linear equations and performing matrix operations effectively.

Row Interpretation of a Matrix

When considering the row interpretation, each row of the matrix represents a linear equation. For a matrix $A$ of size $m \times n$ (where $m$ is the number of rows and $n$ is the number of columns), the matrix can be written out as:

A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}

Linear Equations Representation

In row interpretation, each row can be associated with a linear equation. For example, the first row of the matrix $A$ :

\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \end{bmatrix}

Can be expressed as a linear equation:

a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = b_1

Where $b_1$ is a value from a column matrix (vector) $B$ representing the constants on the right-hand side of the equations. Similarly, the $i$ -th row of $A$ corresponds to:

a_{i1} x_1 + a_{i2} x_2 + \cdots + a_{in} x_n = b_i

System of Linear Equations

Therefore, a matrix equation $A \mathbf{x} = \mathbf{b}$ can be interpreted in terms of rows, where each row corresponds to one equation in a system of $m$ linear equations with $n$ unknowns. Here $\mathbf{x}$ is the column vector of variables:

\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}

And $\mathbf{b}$ is the column vector of constants:

\mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}

Practical Applications

Understanding the row interpretation of matrices is essential in fields such as:

Linear Algebra: To solve systems of linear equations using methods like Gaussian elimination.
Computer Graphics: To perform transformations (translation, rotation, scaling) using transformation matrices.
Data Analysis: For operations like regression analysis, principal component analysis (PCA), and more.

Summary

In summary, the row interpretation of matrices involves visualizing each row of a matrix as a separate linear equation, which together form a system of equations. This perspective is fundamental in solving various mathematical and real-world problems efficiently.