Question

Solve by determinants. Evaluate the determinants by calculator or by minors. $$\begin{aligned}&2 x-y-z-w=0\\\&x-3 y+z+w=0\\\&x+y-4 z+w=0\\\&x+y+w=36\end{aligned}$$

Short answer

To solve the given system of equations by determinants: First, evaluate the determinant of the coefficient matrix (\( \text{det}(A) \)) and the determinants of the matrices obtained by replacing each column of \( A \) with the constant matrix (\( \text{det}(A_i) \)). Then, apply Cramer's Rule to find the values for \( x, y, z, \) and \( w \).

Step-by-step solution

Set up the coefficient matrix

Write the coefficients of the variables in matrix form. Be sure to include the coefficients for all variables, including the constants on the other side of the equals sign from the variables. For the system of equations given, the coefficient matrix is \[ A = \begin{bmatrix} 2 & -1 & -1 & -1 \ 1 & -3 & 1 & 1 \ 1 & 1 & -4 & 1 \ 1 & 1 & 0 & 1 \end{bmatrix} \]

Set up the constant matrix

Write the matrix of constants, which are the values on the other side of the equal signs. In this case, the constants are all zeroes except for the last equation. So we have \[ B = \begin{bmatrix} 0 \ 0 \ 0 \ 36 \end{bmatrix} \]

Compute the determinants

Using a calculator or the method of minors, calculate the determinant of the coefficient matrix, \( \text{det}(A) \), and determinants of matrices formed by replacing each column of \( A \) with the constant matrix \( B \), one at a time. These determinants are denoted \( \text{det}(A_i) \), where \( i \) indicates which column is replaced.

Apply Cramer's Rule

To find the values of \( x, y, z, \) and \( w \), apply Cramer's Rule which states that if \( \text{det}(A) eq 0 \), each variable is found by dividing the determinant of the matrix with its column replaced by \( B \) by the determinant of the coefficient matrix, i.e., \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \), \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \), \( z = \frac{\text{det}(A_z)}{\text{det}(A)} \), and \( w = \frac{\text{det}(A_w)}{\text{det}(A)} \).

Solve for the variables

After calculating all the determinants, divide them accordingly to find the values of \( x, y, z, \) and \( w \). Write down these solutions to conclude the process.

Key concepts

Determinants

Determinants are a fundamental concept used in linear algebra to analyze the properties of matrices. They are scalar values that summarize certain properties of a matrix, particularly whether a matrix is invertible or not. When faced with a square matrix, such as the coefficient matrix in a system of linear equations, the determinant can tell us if the system has a unique solution. This is crucial because it affects how we can approach solving the system.

Calculating the determinant of a matrix involves a series of operations that include multiplication, addition, and subtraction. For larger matrices, this process is typically done by breaking down the matrix using various methods, such as expansion by minors or using more advanced algorithms for efficiency. In the context of Cramer's Rule, the determinant of the coefficient matrix must be non-zero to ensure the system of equations has a unique solution, allowing us to use the rule effectively.

System of Linear Equations

A system of linear equations is a collection of equations that are all linear. These equations are related by sharing the same set of variables, and the solution to the system is the set of values for these variables that satisfies all the equations simultaneously. In the context of the given exercise, we are looking at a system of four linear equations with four variables: x, y, z, and w. Solving this type of system can be approached in different ways, including graphically, by substitution, elimination, or using matrix methods like Cramer's Rule when applicable.

A key requirement for Cramer's Rule is that the system must be defined with the same number of equations as variables, and the determinant of the coefficient matrix must not be zero. If those conditions are met, the rule provides a direct path to find the values of each variable.

Coefficient Matrix

In the realm of linear algebra, the coefficient matrix is a key construct when dealing with systems of linear equations. It is simply a matrix formed by lining up the coefficients of the variables from a system. This matrix plays a central role in various methods for solving systems, such as matrix inversion and Cramer's Rule. In the given example, the coefficient matrix A is a 4x4 matrix that encapsulates all the information about the system's variables coefficients.

To apply Cramer's Rule, one needs to understand how to set up this matrix properly. Each row in the matrix corresponds to an equation, while each column corresponds to a variable. Importantly, the order of both equations and variables should be consistent to prevent errors in the solution process.

Method of Minors

The method of minors is a technique used to calculate the determinant of a matrix, especially when dealing with larger matrices. This approach involves finding the 'minor' for each element of the matrix which is a determinant of a smaller matrix created by removing the row and column of the element in question. Then one has to consider the sign associated with the element's position, leading to the concept of 'cofactors'.

When applying the method of minors, one creates a matrix of cofactors and then uses these to expand along a row or column to find the determinant. For instance, in a 4x4 matrix, the minor of a particular element is the determinant of the 3x3 matrix that remains after removing the element's corresponding row and column. This method was mentioned in the step-by-step solution as a potential way to calculate the determinant needed to apply Cramer's Rule.