Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable". \(\left\\{\begin{array}{r}x-2 y+3 z=0 \\ 3 x+y-2 z=0 \\ 2 x-4 y+6 z=0\end{array}\right.\)

Short answer

Not applicable

Step-by-step solution

Determine the coefficient matrix (A)

Identify the coefficients of the variables from each equation to form matrix A.\[ A = \begin{pmatrix} 1 & -2 & 3 \ 3 & 1 & -2 \ 2 & -4 & 6 \end{pmatrix} \]

Compute the determinant of the coefficient matrix (det(A))

To check if Cramer's Rule is applicable, calculate the determinant of matrix A.\[ \text{det}(A) = \begin{vmatrix} 1 & -2 & 3 \ 3 & 1 & -2 \ 2 & -4 & 6 \end{vmatrix} \]Expanding along the first row,\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 1 & -2 \ -4 & 6 \end{vmatrix} - (-2) \cdot \begin{vmatrix} 3 & -2 \ 2 & 6 \end{vmatrix} + 3 \cdot \begin{vmatrix} 3 & 1 \ 2 & -4 \end{vmatrix} \]Calculating the sub-determinants,\[ \begin{vmatrix} 1 & -2 \ -4 & 6 \end{vmatrix} = (1)(6) - (-2)(-4) = 6 - 8 = -2 \]\[ \begin{vmatrix} 3 & -2 \ 2 & 6 \end{vmatrix} = (3)(6) - (-2)(2) = 18 + 4 = 22 \]\[ \begin{vmatrix} 3 & 1 \ 2 & -4 \end{vmatrix} = (3)(-4) - (1)(2) = -12 - 2 = -14 \]Substitute back into the determinant equation,\[ \text{det}(A) = 1 \cdot (-2) - (-2) \cdot 22 + 3 \cdot (-14) \= -2 + 44 - 42 = 0 \]

Analyze the determinant

Since \( \text{det}(A) = 0\), the coefficient matrix A is singular, meaning it does not have an inverse. Therefore, Cramer's Rule is not applicable in this case.

Key concepts

Determinant of a Matrix

In the world of linear algebra, the **determinant** is a special number that can be calculated from a square matrix. It provides important properties about the matrix and the system of linear equations it represents. Given a matrix A, the determinant is denoted as \( \text{det}(A) \) or sometimes just |A|.

The determinant is calculated through a specific arithmetic process. For a 3x3 matrix like the one in the original exercise, the calculation involves decomposing the matrix into smaller 2x2 determinants (sub-determinants). Each sub-determinant is found by removing one row and one column and then applying the rule: \( ad - bc \).

Therefore, if you have a 3x3 matrix:

• Expand along any row or column
• Compute the sub-determinants
• Add and subtract them accordingly

In practice, this helps determine certain properties, such as whether a system of equations has a unique solution, infinitely many solutions, or no solution.

System of Linear Equations

A **system of linear equations** is a collection of one or more linear equations involving the same set of variables. For example, in the original exercise:

• \( x - 2y + 3z = 0 \)
• \( 3x + y - 2z = 0 \)
• \( 2x - 4y + 6z = 0 \)

We aim to find values of x, y, and z that satisfy all these equations simultaneously.

There are three possible outcomes when solving such systems:

• **Unique Solution:** One specific set of values solves all equations (this happens when the determinant of the coefficient matrix is non-zero).
• **No Solution:** No sets of values satisfy all equations (usually happens when the equations represent parallel lines).
• **Infinitely Many Solutions:** There are multiple (or infinite) sets of values that can solve the equations (this often occurs when the determinant is zero, indicating dependent equations).

Cramer's Rule is a method to solve for each variable when the determinant is non-zero, ensuring a unique solution exists.

Singular Matrix

A **singular matrix** is a square matrix that does not have an inverse. This is indicated by its determinant being zero (\( \text{det}(A) = 0 \)).

When dealing with systems of linear equations, a singular matrix means that the system either has no solution or infinitely many solutions, but not a unique solution. This is the conclusion we reached in the original exercise when we found that \( \text{det}(A) = 0 \).

Some key points to remember about singular matrices:

• They cannot be inverted.
• They indicate linear dependence within the rows or columns, meaning some rows or columns are just multiples of others.
• In the context of solving linear equations, they tell us that Cramer's Rule is not applicable.

Understanding the properties of singular matrices is crucial in determining the behavior of systems of equations and identifying how many solutions are present.