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Chapter 6: Problem 59

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's Rule.

Short Answer

Expert verified
To use Cramer's rule in solving a linear system with three variables, write down the system and form a coefficient matrix. Calculate the determinant of this matrix. Create minor matrices by replacing one column at a time with the constants from the equations. Calculate the determinant for each minor matrix, then solve for each variable by dividing the determinant of the respective minor matrix by the determinant of the coefficient matrix.

Step by step solution

01

Write down the system

To start with, write down the system of equations that is to be solved. Supposing the system has the form: \(ax + by + cz = j\), \(dx + ey + fz = k\), and \(gx + hy + iz = l\)
02

Write down the coefficient matrix

The coefficient matrix \(A\) is formed by the coefficients of the variables in the system of equations. In this case it is \[A= \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]
03

Calculate the determinant of A

The determinant \(D\) is calculated by the coefficients of the variables from the system of equations. Use the determinant formula for a 3x3 matrix.
04

Minor Matrices, or Variable Matrices

We'll need to create new matrices by substituting our answers \(j, k, l\), one column at a time.
05

Calculate the determinants

For each of the matrices from step 4, calculate the determinant. Be careful to substitute in the correct column depending on the variable being solved for
06

Solve for the variables using Cramer's Rule

Using Cramer's Rule, each of the variable \(x, y, z\) is given as the ratio of the determinant of the respective minor matrix to the determinant of the coefficient matrix \(D\). For example, \( x = \frac{D_x}{D}, y = \frac{D_y}{D}, z = \frac{D_z}{D} \)

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Most popular questions from this chapter

When expanding a determinant by minors, when is it necessary to supply minus signs?

Exercises \(85-87\) will help you prepare for the material covered in the next section. Multiply: $$\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} &a_{22}\end{array}\right]\left[\begin{array}{ll}1 & 0 \\\0 & 1\end{array}\right]$$ After performing the multiplication, describe what happens to the elements in the first matrix.

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrr}-2 & 1 & -1 \\\\-5 & 2 & -1 \\\3 & -1 & 1\end{array}\right]$$

Explain how to solve the matrix equation \(A X=B\).

What is meant by the order of a matrix? Give an example with your explanation.
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