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Chapter 6: Problem 52

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

Short Answer

Expert verified
To write a system of linear equations as a matrix equation, place the coefficients of the equations in a matrix \(A\), the variables in a vector \(X\), and the constants in a vector \(B\). The resulting matrix equation is \(AX = B\).

Step by step solution

01

Definition of Linear system of equations

A linear system of three equations in three variables is a set of equations of the form: \(ax + by + cz = d\) \(ex + fy + gz = h\) \(ix + jy + kz = l\) where \(x, y, z\) are the variables, and \(a, b, c, d, e, f, g, h, i, j, k, l\) are coefficients.
02

Matrix representation

The three equations can be rewritten as a single matrix equation, \(AX = B\), where \(A\) = \(\begin{bmatrix} a & b & c \ e & f & g \ i & j & k \end{bmatrix}\), \(X\) = \(\begin{bmatrix} x \ y \ z \end{bmatrix}\) and \(B\) = \(\begin{bmatrix} d \ h \ l \end{bmatrix}\).
03

Formulation of Matrix Equation

Thus, the matrix equation representing the system of equations would be, \(\begin{bmatrix} a & b & c \ e & f & g \ i & j & k \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} d \ h \ l \end{bmatrix}\).

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Most popular questions from this chapter

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&3 x=2-3 y\\\&2 y=3-2 x\end{aligned} $$

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{ll} 4 & 2 \\ 6 & 1 \\ 3 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 4 \\ -1 & -2 & 0 \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rrr}2 & -4 & 2 \\\\-1 & 0 & 5 \\\3 & 0 & 4\end{array}\right| $$

The process of solving a linear system in three variables using Cramer's rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

Evaluate: $$\left|\begin{array}{lllll}2 & 0 & 0 & 0 & 0 \\\0 & 3 & 0 & 0 & 0 \\\0 & 0 & 2 & 0 & 0 \\\0 & 0 & 0 & 1 & 0 \\\0 & 0 & 0 & 0 & 4\end{array}\right|$$
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