Question

Solve by determinants. Evaluate the determinants by calculator or by minors. $$\begin{aligned}&x+y+z=35\\\&x-2 y+3 z=15\\\&y-x+z=-5\end{aligned}$$

Short answer

x = det(C_x) / det(A), y = det(C_y) / det(A), z = det(C_z) / det(A), where C_x, C_y, and C_z are the matrices with replaced columns and A is the coefficient matrix.

Step-by-step solution

- Set up the coefficient matrix

Write the coefficient matrix A using the coefficients of the variables x, y, and z from the system of equations.

- Set up the constant matrix

Write the constant matrix (column vector) B using the constants from the right side of the equations.

- Compute the determinant of matrix A

Calculate the determinant of the coefficient matrix A, denoted as det(A), using a calculator or by the method of minors.

- Find determinant of matrix C_x

Create matrix C_x by replacing the first column of A with the constant matrix B. Compute the determinant of C_x, denoted as det(C_x), using a calculator or by minors.

- Find determinant of matrix C_y

Create matrix C_y by replacing the second column of A with B. Calculate the determinant of C_y, denoted as det(C_y), with a calculator or by minors.

- Find determinant of matrix C_z

Create matrix C_z by replacing the third column of A with B. Compute the determinant of C_z, denoted as det(C_z), with a calculator or by minors.

- Solve for x, y, and z

Calculate the values of x, y, and z using Cramer's Rule: x = det(C_x) / det(A), y = det(C_y) / det(A), z = det(C_z) / det(A).

Key concepts

Determinants

A determinant is a special numerical value that can be computed from a square matrix. It is a useful tool in linear algebra, especially when it comes to solving systems of linear equations. In the context of Cramer's Rule, the determinant helps us understand if a system has a unique solution, and if so, what that solution is.

To compute a determinant of a 2x2 matrix, you would take the product of the diagonal elements and subtract the product of the off-diagonal elements. For a 3x3 matrix or higher, the process involves a series of subtractions and multiplications based on the elements of the matrix, or what are called 'minors'. This can be more complex and sometimes calls for simplification of the matrix or the use of technology, like a calculator.

Importantly, if the determinant of the coefficient matrix in a system of linear equations is zero, it indicates that the system has either no solution or an infinite number of solutions. If it's non-zero, it suggests a unique solution, paving the way for Cramer's Rule to be used.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For example, a system could have two equations with two unknowns, or three equations with three unknowns, like in our exercise. Solving these systems involves finding values for the variables that satisfy all equations simultaneously.

Solving systems can be approached in several ways, including graphically, by substitution, elimination, and through matrix operations such as using determinants which is embodied in Cramer's Rule. A system of equations might have a single unique solution, no solution, or infinitely many solutions. The determinant of the coefficient matrix (which we get from the left side of the equations) is crucial in telling us which is the case.

Coefficient Matrix

The coefficient matrix is a square matrix made up of coefficients from a system of linear equations. Each row in the matrix corresponds to an equation, and each column corresponds to a variable. In our previous exercise, the coefficient matrix A is constructed from the coefficients of x, y, and z in the system.

When we compute the determinant of this matrix, we get a value indicative of the system's nature. If this determinant is not zero, we can proceed to use Cramer's Rule to find the solution. When creating matrices to solve systems of equations, precision in placing coefficients into their correct positions within the matrix is critical for an accurate solution.

Matrix Calculations

Matrix calculations can seem daunting, but they are systematic and follow set rules. These calculations include addition, subtraction, multiplication, and finding inverses and determinants of matrices. When solving systems of equations, we're most interested in the calculation of determinants and matrix multiplication.

In Cramer's Rule, after finding the determinant of the coefficient matrix, matrices C_x, C_y, and C_z are formed by replacing the respective columns of the coefficient matrix with the constants from the equations. Calculating the determinants of these new matrices then allows us to solve for the unknown variables.

Understanding how to perform these matrix calculations is essential, as they are the engine behind solving a system of equations through Cramer's Rule. Always ensure the correctness of each step, as a single error can affect the final outcome.