Question

A matrix \(A\) and vector \(\vec{b}\) are given. Solve the equation \(A \vec{x}=\vec{b},\) write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane. $$ A=\left[\begin{array}{cc} 2 & -5 \\ -4 & -10 \end{array}\right], \vec{b}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$

Short answer

The solution is \(\vec{x} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\). Sketch shows only the origin.

Step-by-step solution

Analyze the System

The system of equations can be represented as follows:\[\begin{align*}2x - 5y &= 0, \-4x - 10y &= 0.\end{align*}\]This system is homogeneous due to \(\vec{b} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\).

Examine the Matrix

Calculate the determinant of matrix \(A\) to determine the nature of the solution. The determinant \(\text{det}(A)\) is given by:\[\text{det}(A) = (2)(-10) - (-5)(-4) = -20 - 20 = -40\]Since the determinant is non-zero, the matrix is not singular.

Find the Solution

Since the determinant is non-zero, the only solution to the homogeneous system is the trivial solution. Thus, the solution is: \[\vec{x} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]

Sketch the Solution

For a homogeneous system with only the zero solution, there is no specific line or plane to be sketched as a result of the equations. The solution point is simply the origin (0, 0) on the Cartesian plane.

Key concepts

Determinant of a Matrix

When you encounter a matrix in a system of equations, understanding its determinant is vital. The determinant, denoted by \( \text{det}(A) \), is a special number that can tell us a lot about the matrix and the system it represents. For a 2x2 matrix\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The determinant is calculated as:\[\text{det}(A) = ad - bc.\]In the context of systems of equations, if the determinant is zero, the matrix does not have an inverse, and the system is said to be singular. However, if the determinant is non-zero, the matrix is non-singular, and the system has a unique solution.
In our exercise, the calculation of the determinant \(\text{det}(A) = -40 \) confirms that matrix \(A\) is non-singular, implying a unique trivial solution for this homogeneous equation.

Trivial Solution

In the realm of linear equations, a 'trivial solution' refers to a solution where all variables are equal to zero. It appears in homogeneous systems, which are systems where all terms not involving the variables are zero, like our given system \( A\vec{x} = \vec{b} \) with \( \vec{b} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \).
The trivial solution is always present in homogeneous equations. It represents the idea that doing nothing to all variables still satisfies the equation. For example, if we have \( 2x - 5y = 0 \) and \( -4x - 10y = 0 \), setting \( x=0 \) and \( y=0 \) solves both equations. In the exercise, since the determinant of \( A \) is non-zero, this trivial solution (\( \vec{x} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \)) is the only one.

Singular vs Non-singular Matrix

The concepts of singular and non-singular matrices stem from linear algebra, relating to whether a matrix has an inverse. If a matrix is singular, it implies it does not have an inverse and its determinant is zero. In contrast, a non-singular matrix has a non-zero determinant and thus possesses an inverse.
This distinction is crucial, especially when solving linear systems:

• **Singular Matrix:** Often leads to no solution or infinitely many solutions due to linearly dependent rows or columns.
• **Non-singular Matrix:** Typically results in a unique solution because the matrix has independent rows or columns.

In this exercise, the matrix \( A \) had a non-zero determinant (\(-40\)), confirming it was non-singular. This ensured there was a unique solution to the homogeneous equation, specifically the trivial solution.