Problem 38 - College Algebra

Question

a. Write each linear system as a matrix equation in the form \(A X=B\) b. Solve the system using the inverse that is given for the coefficient matrix. $$ \begin{array}{r} x+2 y+5 z=2 \\ 2 x+3 y+8 z=3 \\ -x+y+2 z=3 \end{array} $$ The inverse of $$ \left[\begin{array}{rrr} 1 & 2 & 5 \\ 2 & 3 & 8 \\ -1 & 1 & 2 \end{array}\right] \text { is }\left[\begin{array}{rrr} 2 & 1 & 1 \\ 12 & 7 & 2 \\ 5 & 3 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution of the system of equations is \(x = 7, y = 39, z = 16\).

Step 1: Rewrite as matrix equation

The system of equations can be written as the matrix equation \(AX = B\). In this equation, matrix A represents the coefficients of x, y and z, X represents the column matrix of x, y, and z, and B represents the column matrix on the right hand side of the equations. Therefore, \(A = \left[\begin{array}{rrr} 1 & 2 & 5 \ 2 & 3 & 8 \ -1 & 1 & 2 \end{array}\right]\), \(X = \left[\begin{array}{rrr} x \ y \ z \end{array}\right]\), and \(B = \left[\begin{array}{rrr} 2 \ 3 \ 3 \end{array}\right]\).

Step 2: Solve using inverse of matrix A

Using the given inverse of matrix A, denoted by \(A^{-1}\), the solution of the system can be found by multiplying \(A^{-1}\) with the matrix B, i.e., \(X = A^{-1}B\). Given that the inverse of matrix A is \( \left[\begin{array}{rrr} 2 & 1 & 1 \ 12 & 7 & 2 \ 5 & 3 & 1 \end{array}\right]\), we can substitute for \(A^{-1}\) and \(B\), and perform matrix multiplication to get the values of x, y and z.

Step 3: Matrix Multiplication

Performing the multiplication, \( \left[\begin{array}{rrr} x \ y \ z \end{array}\right] = \left[\begin{array}{r} 2(2) + 1(3) + 1(3) \ 12(2) + 7(3) + 2(3) \ 5(2) + 3(3) + 1(3) \end{array}\right] = \left[\begin{array}{r} 7 \ 39 \ 16 \end{array}\right]\). Therefore, the solutions are \(x = 7, y = 39, z = 16\).

Key Concepts

Matrix Multiplication

Matrix multiplication is one of the fundamental operations in linear algebra. It's how we combine two matrices to produce a third matrix. Unlike scalar multiplication, where we simply multiply two numbers, matrix multiplication involves a specific ruleset for combining values.

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. During the multiplication process, we take the rows from the first matrix and columns from the second, performing multiplication and addition as follows: Each element of the resulting matrix is the sum of the products of corresponding elements in the row of the first and column of the second matrix.

For example, if we are multiplying a matrix A with dimensions 2x3 (2 rows and 3 columns) and a matrix B with dimensions 3x2 (3 rows and 2 columns), the resulting matrix will have dimensions of 2x2 (2 rows and 2 columns). Here's a simplified visual representation:

• Resulting matrix C[1,1] = Sum of (A[1,k] * B[k,1]) for all k
• Resulting matrix C[1,2] = Sum of (A[1,k] * B[k,2]) for all k
• ... and so on for each element of matrix C.

This principle is crucial for solving systems of equations as we translate these into matrix operations for more efficient calculations.

Matrix Inverse

A matrix inverse is particularly powerful in solving systems of linear equations. Analogous to the reciprocal of a number, the inverse of a matrix A, denoted as \(A^{-1}\), is a unique matrix that, when multiplied with the original, results in the identity matrix. The identity matrix is the equivalent of the number '1' in matrix form, where all the elements on the main diagonal are 1, and all other elements are 0.

However, not every matrix has an inverse. Some matrices, known as singular matrices, do not have inverses. For a matrix to have an inverse, it must be square (same number of rows and columns) and its determinant must not be zero. Finding the inverse is a process of performing several algebraic steps, including finding the determinant, the matrix of minors, the matrix of cofactors, and the adjugate, and finally dividing by the determinant.

In the context of solving systems of equations, once we have the inverse of the coefficient matrix A, represented as \(A^{-1}\), we can multiply both sides of the matrix equation \(AX = B\) by \(A^{-1}\) to find \(X\), where \(X\) is the matrix representing the variables of the system.

Matrix Equation Form

The matrix equation form is a convenient and compact way to represent a system of linear equations. This form translates the system into matrices and vectors, which allows for efficient computations using matrix operations. A system of linear equations can be expressed as \(AX = B\), where:

• \(A\) is the coefficient matrix,
• \(X\) is the column matrix (or vector) containing the variables,
• \(B\) is the column matrix (or vector) of constants.

It's important to arrange the elements of the matrices correctly to preserve the relationships inherent in the system of equations. This arrangement ensures that when we apply operations like matrix multiplication or inversion, they correspond correctly to operations we would perform on the linear system.

For instance, in our example, three equations with variables x, y, and z are reformulated into a matrix equation using \(A\), \(X\), and \(B\), which streamlines the process of finding the variables' values by leveraging matrix operations instead of dealing with the equations individually.

Linear Algebra

Linear algebra is a branch of mathematics that is pivotal in modern computation and engineering, particularly for solving systems of linear equations. This area of study focuses on vectors, vector spaces, linear transformations, and matrices. The strength of linear algebra lies in its ability to simplify complex problems into matrix and vector operations that can be solved systematically.

In linear algebra, a system of linear equations can be solved in various ways, including graphical methods, substitution, elimination, and matrix operations. These methods are applicable depending on the nature and complexity of the system. The use of matrices and matrix operations, as described earlier, often provides the most straightforward and computationally efficient approach, especially for larger systems.

The concepts of matrix multiplication, inversion, and equation form are foundational in this field, making it easier to analyze and solve large-scale problems that otherwise would be too daunting to approach with basic algebra techniques alone. Understanding these concepts is essential for anyone delving into areas that require solving systems of linear equations, whether in mathematics, science, engineering, or computer science.