Question

Solve by determinants. Evaluate the determinants by calculator or by minors. $$\begin{aligned}&3 x-2 y-z+w=-3\\\&-x-y+3 z+2 w=23\\\&x+3 y-2 z+w=-12\\\&2 x-y-z-3 w=-22\end{aligned}$$

Short answer

After calculating the determinants using either a calculator or the method of minors, divide the determinant of each modified matrix by the determinant of the coefficient matrix to find the values of x, y, z, and w.

Step-by-step solution

Set up the matrix for the system of equations

Write the system of linear equations as an augmented matrix to prepare for solving by determinants. In each row, the coefficients of variables x, y, z, and w correspond to the columns, and the last column is the constant from the right-hand side of each equation.

Find the determinant of the coefficient matrix

Calculate the determinant of the matrix formed by the coefficients of the variables. The matrix is a 4x4 matrix since there are four equations and four variables.

Apply Cramer's Rule to find the variables

Use Cramer's Rule by substituting each column of the coefficient matrix with the constants from the right side of the equation one at a time and calculate each determinant to find the values of x, y, z, and w.

Calculate determinants for each variable

For each variable, replace the corresponding column of the coefficient matrix with the constants and calculate the determinant of this new matrix.

Divide to find the solutions

The solution for each variable will be the determinant found in Step 4 divided by the determinant found in Step 2. This will give the values of x, y, z, and w.

Key concepts

Determinants

Determinants play a crucial role in solving systems of linear equations, especially when dealing with multiple variables. Think of a determinant as a special number that can be calculated from a square matrix. In the context of our exercise, the determinant helps us understand whether a unique solution exists for our system of equations.

For a 2x2 matrix, the determinant is calculated by subtracting the product of the top-right and bottom-left entries from the product of the top-left and bottom-right entries. For larger matrices, like the 4x4 matrix in our exercise, the process is more complex and usually involves breaking down the matrix into smaller 'minors' and using a method called 'cofactor expansion'. Calculators or computer software can significantly speed up this process.

It's important to note that if the determinant of the coefficient matrix is zero, the system of equations is either dependent or inconsistent and does not have a unique solution. However, if the determinant is non-zero, Cramer's Rule can be applied to find a unique solution.

Cramer's Rule

Cramer's Rule is a theorem that gives an explicit formula for the solution of a system of linear equations with as many equations as unknowns, provided that the system has a unique solution. This rule uses the determinants of the coefficient matrix and matrices obtained from it by replacing one column by the column of constants.

To apply Cramer's Rule, first ensure that the determinant of the coefficient matrix is non-zero. Then, for each variable, construct a new matrix by replacing the column corresponding to that variable with the constants from the right-hand sides of the equations. The value of the variable is found by dividing the determinant of this new matrix by the determinant of the coefficient matrix. This process must be repeated for each variable. It's an elegant solution method, but it becomes computationally heavy for systems with a large number of variables.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It includes both the coefficients of the variables and the constants from the right-hand sides of the equations in a single matrix. Typically, it's written as a block matrix with a line (usually a vertical bar) separating the coefficient matrix on the left from the constants on the right.

In our exercise, the augmented matrix would consist of a 4x5 matrix where the first four columns represent the coefficients of the variables x, y, z, and w, and the fifth column represents the constants -3, 23, -12, and -22. This representation is particularly useful because it allows us to perform row operations to solve the system or to set up for methods like determinant-based Cramer's Rule.

Coefficient Matrix

The coefficient matrix is a key component when solving systems of equations algebraically, especially with methods like Cramer's Rule. It's simply a matrix formed by organizing the coefficients of the variables from a system of linear equations into rows and columns. Each row represents an equation, and each column corresponds to a variable.

In the exercise, the coefficient matrix is a 4x4 matrix because we have four equations and four unknowns (x, y, z, and w). This matrix does not include the constants from the equations; those are kept separate in the augmented matrix. The determinant of the coefficient matrix is the denominator in the computation for each variable when applying Cramer's Rule. It's critical for this determinant to be non-zero to proceed with finding a unique solution for the system of equations.