Problem 32 - Precalculus Mathematics for Calculus

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 is -46; (c) No other solutions; (d) Yes, Cramer's Rule applies.

Step 1: Verify the Given Solution

Substitute the values of \(x = -1\), \(y = 0\), and \(z = 1\) into each equation of the system. For the first equation: \(x + 2y + 6z = 5\), substitute to get: \(-1 + 2(0) + 6(1) = -1 + 6 = 5\). The first equation holds true.For the second equation: \(-3x - 6y + 5z = 8\), substitute to get: \(-3(-1) - 6(0) + 5(1) = 3 + 5 = 8\). The second equation holds true.For the third equation: \(2x + 6y + 9z = 7\), substitute to get: \(2(-1) + 6(0) + 9(1) = -2 + 9 = 7\). The third equation holds true.Thus, the given values are a solution to the system.

Step 2: Find the Determinant of the Coefficient Matrix

The coefficient matrix is:\[\begin{bmatrix}1 & 2 & 6 \-3 & -6 & 5 \2 & 6 & 9\end{bmatrix}\]The determinant can be found using the formula for a 3x3 matrix:\[\begin{vmatrix}a & b & c \d & e & f \g & h & i\end{vmatrix}= a(ei − fh) − b(di − fg) + c(dh − eg)\]Substitute the values: \(a = 1, b = 2, c = 6, d = -3, e = -6, f = 5, g = 2, h = 6, i = 9\).Calculate:\[1((-6)(9) - (5)(6)) - 2((-3)(9) - (5)(2)) + 6((-3)(6) - (-6)(2))\]\[= 1(-54 - 30) - 2(-27 - 10) + 6(-18 + 12)\]\[= 1(-84) + 2(37) + 6(-6)\]\[= -84 + 74 - 36 = -46\]The determinant is \(-46\).

Step 3: Determine Other Solutions

Since the determinant of the coefficient matrix is non-zero (i.e., \(-46\)), the system of equations has a unique solution. Hence, there are no other solutions.

Step 4: Can Cramer's Rule be Used?

Cramer's Rule can be used to solve a system of equations only if the determinant of the coefficient matrix is non-zero. Since the determinant is \(-46\), which is non-zero, Cramer's Rule can be applied to solve this system of equations.

Key Concepts

Determinant of a Matrix

In the realm of linear algebra, the determinant of a matrix is a special number that can be calculated from the square matrix. For a 3x3 matrix, the calculation of the determinant is particularly important as it helps in determining the nature of the solution. The process involves a specific formula: if matrix \\[\begin{bmatrix} a & b & c \d & e & f \g & h & i \end{bmatrix}\]Then, the determinant \can be calculated using the formula:\[det = a(ei - fh) - b(di - fg) + c(dh - eg).\]
Calculating the determinant tells us if the matrix is invertible. If the determinant is zero, the matrix is not invertible and hence the system of equations may have no solution or infinitely many solutions. However, if the determinant is non-zero, as in the case of our example with the determinant \\(-46\), the matrix has an inverse, indicating a potentially unique solution.

Cramer's Rule

Cramer's Rule is a powerful tool for solving systems of linear equations using determinants. It's especially useful when the coefficient matrix of the system has a non-zero determinant. The rule states that if you have a system of equations with the same number of equations as unknowns (like our 3x3 system), you can solve it using determinants.

The solution to each variable can be found by substituting the column of that variable with the constants from the equations on the right-hand side and then dividing this new determinant by the determinant of the coefficient matrix. This method provides a straightforward way to find the solution, especially when the determinant of the coefficient matrix is non-zero. Since the determinant in our example is \\(-46\), Cramer's Rule can indeed be applied here.

Unique Solution

A unique solution in a system of linear equations means there is exactly one set of values for the variables that satisfies all the equations in the system. When you calculate the determinant of the coefficient matrix and find it to be non-zero, like we did with \\(-46\), you confirm the system has a unique solution.

This is a critical concept as it reassures us that the system of equations is consistent and independent. Consistent means that the equations do not contradict each other, while independent means each equation provides new information about the system. Hence, a non-zero determinant is a foolproof indicator that there is one and only one solution possible.

Coefficient Matrix

The coefficient matrix is derived by extracting the coefficients of the variables from a system of linear equations. In our exercise, it is formed by the constants multiplying the variables in all the equations:\[\begin{bmatrix} 1 & 2 & 6 \-3 & -6 & 5 \2 & 6 & 9 \end{bmatrix}\]. This matrix is central to solving the system using matrix methods such as finding the determinant or applying Cramer's Rule.

A well-formed coefficient matrix provides the necessary structure for exploring the solutions of the system. By focusing on the coefficients, we can analyze the dependencies and relationships between the equations. This matrix allows mathematicians to employ techniques like row operations, determinant evaluation, and inverse calculations, enabling deeper insight into the system's characteristics and solutions.