Problem 57 - Precalculus with Limits

Question

In Exercises 51-58, use an inverse matrix to solve (if possible) the system of linear equations. \(\begin{cases} 4x - y + z = -5 \\ 2x + 2y + 3z = 10 \\ 5x - 2y + 6z = 1 \end{cases}\)

Step-by-Step Solution

Answer

The short answer will be the value of x, y, z that we calculated in step 4.

Step 1: Formulate the coefficient and constant matrices

The coefficient matrix \(A\) is formed using the coefficients of the variables (x, y, z) in the linear equations, while the constant matrix \(B\) is formed using the constants on the right side of the equations. Hence, \[A = \begin{pmatrix} 4 & -1 & 1 \ 2 & 2 & 3 \ 5 & -2 & 6 \end{pmatrix}, B = \begin{pmatrix} -5 \ 10 \ 1 \end{pmatrix}\]

Step 2: Calculate the inverse of matrix A

To find the inverse of matrix \(A\), denoted as \(A^{-1}\), you need to perform a series of row operations, including row swapping, row multiplication and row addition, such that matrix \(A\) transforms into the identity matrix. If \(A\) is non-singular (determinant not equal to 0), the inverse exists.

Step 3: Apply the inverse to the constant matrix

The solution to the system of equations can be found by multiplying the constant matrix \(B\) with the inverse matrix \(A^{-1}\), that is, \(X = A^{-1}B\) where \(X\) is the matrix containing the variables (x, y, z).

Step 4: Solve for x, y, z

When finding the product \(A^{-1}B\), the results give the solutions to x, y, z in the order respectively.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find the values of the variables that satisfy all the equations simultaneously. In other words, you look for values of variables that make each individual equation in the system true.
Linear equations are equations of the first order. What makes them 'linear' is their lack of exponents on the variables. Instead, these variables typically act as coefficients or constants.

In the provided exercise, the system is represented as:

• Equation 1: \(4x - y + z = -5\)
• Equation 2: \(2x + 2y + 3z = 10\)
• Equation 3: \(5x - 2y + 6z = 1\)

Solving systems of equations like these means finding the common values of \(x\), \(y\), and \(z\) that balance all the equations. This is precisely where an inverse matrix can be handy when attempting to solve such a system.

Matrix Inversion

Matrix inversion is a mathematical process where you find another matrix that when multiplied by the original matrix yields the identity matrix. The identity matrix is like the number 1 in matrix arithmetic.
Inverting a matrix is only possible if the matrix is square (same number of rows and columns) and non-singular (determinant is not zero). For the matrix \(A\) given in the exercise, the inverse matrix is denoted as \(A^{-1}\).

Computing the inverse involves a set algorithmic approach:

• Ensure the matrix is square and calculate its determinant.
• If the determinant is non-zero, employ row operations to convert the matrix to an identity matrix.

Inverse matrices are fundamental when solving linear equations using matrix methods as they help isolate the variable matrix (\(X\)) through multiplication.

Determinant of a Matrix

The determinant of a matrix is a special number that provides important information concerning the matrix's properties. Its calculation is essential before attempting matrix inversion.
The determinant helps us understand whether a matrix is invertible or not. If a matrix has a determinant of zero, it is termed singular, and no inverse can be computed. For our matrix \(A\) in the problem, we check its determinant before attempting any further operations.

Calculating the determinant of a 3x3 matrix involves:

• Expanding the matrix using one of its rows or columns.
• Using the square minors and cofactors.

The determinant is not just a number—it signifies the scale factor of transformation described by the matrix in geometric terms.

Row Operations

Row operations are fundamental techniques in linear algebra used during matrix manipulation and solving systems of equations. There are three basic types of row operations:

• Row swapping: This means exchanging two rows with each other.
• Row multiplication: Where any row in a matrix can be multiplied by a non-zero scalar.
• Row addition: A row can be replaced by adding it to the multiple of another row.

These operations are particularly useful when performing the algorithm to find an inverse matrix. By applying these operations step by step, you can transform any matrix into its reduced row-echelon form or even to the identity matrix.
In our exercise, row operations were crucial in transforming matrix \(A\) to its inverse form. These operations help us systematize the approach to solving linear equations because they maintain the solution set of the system.