Problem 60 - College Algebra

Question

Explain how to write a linear system of three equations in three variables as a matrix equation.

Step-by-Step Solution

Answer

The matrix equation corresponding to a system of linear equations is constructed by forming three matrices: a coefficient matrix (from the coefficients of the variables in the equations), variables matrix (the variables in the equations), and a constants matrix (the constants on the other side of the equals sign in the equations). The resulting matrix equation takes the form \( AX = B \).

Step 1: Understand the components of a matrix equation

A matrix equation is typically of the form \( AX = B \). Here \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constants matrix.

Step 2: Identify the system of equations

Identify the system of equations. Suppose the equations given are: \( a1x + b1y + c1z = d1 \), \( a2x + b2y + c2z = d2 \), \( a3x + b3y + c3z = d3 \).

Step 3: Form the Coefficient Matrix A

Arrange all of the coefficients into matrix A: \( A = \[ [a1, b1, c1], [a2, b2, c2], [a3, b3, c3]] \]. Each row of this matrix represents the coefficients of one of the equations.

Step 4: Form the Variable Matrix X

The variable matrix is a column matrix containing the variables: \( X = \[ [x], [y], [z] ] \).

Step 5: Form the Constant Matrix B

The constants matrix is a column matrix formed from the constants of each equation: \( B = \[ [d1], [d2], [d3] ] \).

Step 6: Write the Matrix Equation

The matrix equation for the given system of linear equations is: \( AX = B \). Substituting A, X, and B from steps 3, 4, and 5, an equation of the form \( \[ [a1, b1, c1], [a2, b2, c2], [a3, b3, c3]] \[ [x], [y], [z] ] \) = \( \[ [d1], [d2], [d3] ] \) is obtained.