Question

Does the system of linear equations have more than one solution? Justify your answer. $$ \begin{aligned} &4 x+y+z=0 \\ &2 x+\frac{1}{2} y-3 z=0 \\ &-x-\frac{1}{4} y-z=0 \end{aligned} $$

Short answer

The given system of linear equations has a unique solution, because the determinant of the coefficient matrix is not equal to zero.

Step-by-step solution

Set Up the Coefficient Matrix

The system of equations can be written in matrix form as \(Ax=B\) where \(A\) is the coefficient matrix. For the given system, the 3 x 3 matrix \(A\) is: \[A = \begin{pmatrix} 4 & 1 & 1\ 2 & 0.5 & -3 \ -1 & -0.25 & -1\end{pmatrix}\] and \(B = \begin{pmatrix} 0 \ 0 \ 0\end{pmatrix}\).

Calculate the Determinant of the Matrix

Use the formula for the determinant of a 3 x 3 matrix to find det(A). det(A) = \(4*(0.5*-1 - (-3*-0.25)) - 1*(2*-1 - 3*-1) + 1*(2*-0.25 - 0.5*-1)\).\nComputing this gives det(A) = 1.

Analyze the Determinant for the Number of Solutions

Since det(A) ≠ 0, this indicates that the system of equations has a unique solution.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that helps us understand some properties about the matrix, especially when dealing with systems of linear equations. For a 3 x 3 matrix like in the exercise, the formula for calculating the determinant is important. The determinant,\[det(A)\], is computed by following a specific formula that involves multiplying and subtracting certain elements of the matrix.

• A non-zero determinant (\(det(A) eq 0\)) tells us that the matrix has full rank and is invertible.
• A zero determinant (\(det(A) = 0\)) means the matrix might be "singular", or not invertible, indicating infinite solutions or no solution exists for the system of equations based on this matrix.

In our exercise, the determinant was calculated as 1, which is not zero. This has specific implications on the solutions of the system.

Unique Solution

A system of linear equations can have no solution, one unique solution, or infinitely many solutions. When the determinant of the coefficient matrix is not zero, like in this exercise where \(det(A) = 1\), the system has a unique solution.

• A non-zero determinant implies that every variable in the system corresponds to exactly one value. There is no overlap or ambiguity, making the solution "unique."
• In practical terms, finding a unique solution means that we can solve the system to find a precise value for each unknown variable in the equations.

Because the determinant was 1, we confidently state that each variable in our given set of equations has only one solution.

Coefficient Matrix

In solving systems of linear equations, the coefficient matrix is crucial. It consists of all the coefficients of the variables from the linear equations arranged in rows and columns.

• For our set of equations, the coefficient matrix is expressed as \(A = \begin{pmatrix} 4 & 1 & 1\ 2 & 0.5 & -3 \ -1 & -0.25 & -1\end{pmatrix}\).
• The structure of this matrix captures the entire system of equations without the constant terms on the right side of each equation.

Understanding the coefficient matrix helps us in computing the determinant and subsequently in determining the number of solutions the system has. It essentially translates a set of equations into matrix form, easing various calculations like finding the determinant or using matrix operations to find solutions. This structural aspect of the matrix allows for systematic approaches in solving complex systems.