Problem 32 - Algebra and Trigonometry

Question

Consider the system $$ \left\\{\begin{aligned} x+2 y+6 z &=5 \\\\-3 x-6 y+5 z &=8 \\ 2 x+6 y+9 z &=7 \end{aligned}\right. $$ (a) Verify that \(x=-1, y=0, z=1\) is a solution of the system. (b) Find the determinant of the coefficient matrix. (c) Without solving the system, determine whether there are any other solutions. (d) Can Cramer's Rule be used to solve this system? Why or why not?

Step-by-Step Solution

Answer

(a) Verified. (b) Determinant: \(-46\). (c) Unique solution exists. (d) Cramer's Rule is applicable.

Step 1: Substitute proposed solution into equations

Check if \(x = -1\), \(y = 0\), and \(z = 1\) satisfies all three equations. Substitute these values into each of the equations to verify.For the first equation: \(-1 + 2(0) + 6(1) = 5\) which simplifies to \(5 = 5\), a true statement.For the second equation: \(-3(-1) - 6(0) + 5(1) = 8\) which simplifies to \(8 = 8\), a true statement.For the third equation: \(2(-1) + 6(0) + 9(1) = 7\) which simplifies to \(7 = 7\), a true statement.

Step 2: Write the coefficient matrix

The coefficient matrix \(A\) is constructed using the coefficients of the variables from the system of equations:\[A = \begin{bmatrix} 1 & 2 & 6 \ -3 & -6 & 5 \ 2 & 6 & 9 \end{bmatrix}\]

Step 3: Calculate the determinant of the coefficient matrix

To find the determinant of matrix \(A\), use the formula for a 3x3 matrix:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Substitute values:\[\text{det}(A) = 1((-6)(9) - (5)(6)) - 2((-3)(9) - (5)(2)) + 6((-3)(6) - (-6)(2))\]Simplifying each term:\[ = 1(-54 - 30) - 2(-27 - 10) + 6(-18 + 12)\]\[ = 1(-84) - 2(-37) + 6(-6)\]\[ = -84 + 74 - 36\]\[ = -46\]

Step 4: Determine uniqueness of solutions

Since the determinant \(\text{det}(A)\) is not zero (\(-46\)), the system has a unique solution according to the theory of linear systems.

Step 5: Find out if Cramer's Rule can be applied

Cramer's Rule can be used if the determinant of the coefficient matrix in a system of linear equations is non-zero. Since \(\text{det}(A) = -46 eq 0\), Cramer's Rule can be applied to solve this system.

Key Concepts

Determinants

The determinant is an important number associated with a square matrix, providing valuable information about the matrix itself. In the context of systems of linear equations, the determinant helps to determine whether a unique solution exists.
For a 3x3 matrix, the determinant can be calculated using a specific formula involving the elements of the matrix. This formula can be written as: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where each letter represents an element of the matrix.
The determinant plays a crucial role: if it equals zero, the system may have either no solutions or infinitely many solutions.

• If the determinant is non-zero, there is a unique solution.
• If it is zero, the system might not have a single, determined solution.

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns. This method employs the calculation of determinants to find the values of the variables.
Cramer's Rule is applicable only when the determinant of the matrix of coefficients is non-zero. This non-zero condition guarantees the existence of a unique solution, making Cramer's Rule an effective method for solving systems that meet this criterion.
To use Cramer's Rule, you replace one column of the coefficient matrix with the constants from the right-hand side of the equations and compute the determinant of this new matrix. You do this for each variable in the system.
For example, in a system with variables \(x\), \(y\), and \(z\), the formulas look like this:

• \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\)
• \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\)
• \(z = \frac{\text{det}(A_z)}{\text{det}(A)}\)

Cramer's Rule offers a straightforward approach when applicable, but remember it only works for systems where the coefficient matrix has a non-zero determinant.

Unique Solution

A unique solution in a system of linear equations is a single set of values for the variables that satisfies all equations in the system simultaneously.
The existence of a unique solution can often be confirmed by examining the determinant of the coefficient matrix. If the determinant is non-zero, the system contains exactly one solution.
This is because a non-zero determinant indicates that the matrix is invertible, which in turn signifies that the system of equations is determined.
Simply put:

• A non-zero determinant means the system has a unique solution.
• A determinant of zero could lead to either no solution or an infinite number of solutions.

In linear algebra, this uniqueness property is a significant aspect as it guarantees the certainty and consistency of the solution in applying mathematical models to real-world problems.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the rules and operations that can be performed on them.
Matrices provide a compact way of organizing data and equations, especially useful in systems of linear equations.
Some core operations in matrix algebra include:

• Matrix addition and subtraction: Similar sized matrices can be added or subtracted by adding or subtracting their corresponding elements.
• Scalar multiplication: Every element of the matrix is multiplied by a single number, or scalar.
• Matrix multiplication: Performed by summing the products of rows and columns, an essential operation for many applications.

The ability to change, combine, and manipulate matrices allows mathematicians and scientists to solve complex systems of equations, illustrate transformations, and handle data with ease.
Matrix algebra simplifies these tasks and underscores the structured manipulation of data in advanced mathematical contexts.