Problem 42 - Algebra and Trigonometry

Question

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned}-2 a \quad\quad+c=& 2 \\ a+2 b-c =9 \\ 3 a+5 b+2 c &=22 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \(a = -1\), \(b = 5\), \(c = 0\).

Step 1: Write the System in Matrix Form

The given system of equations is:\\(-2a + 0b + c = 2\)\\(a + 2b - c = 9\)\\(3a + 5b + 2c = 22\)\Rewriting this in matrix form, we have AX = B where A is the coefficient matrix:\\[A = \begin{bmatrix} -2 & 0 & 1 \ 1 & 2 & -1 \ 3 & 5 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} a \ b \ c \end{bmatrix}, \quad B = \begin{bmatrix} 2 \ 9 \ 22 \end{bmatrix}\]

Step 2: Calculate Determinant of A

The determinant of matrix A, \(\text{det}(A)\), is calculated as:\\[\text{det}(A) = \begin{vmatrix} -2 & 0 & 1 \ 1 & 2 & -1 \ 3 & 5 & 2 \end{vmatrix}\] \[= (-2)(2 \times 2 + 1 \times 5) - 0(1 \times 2 - 1 \times 3) + 1(1 \times 5 - 2 \times 3)\] \[= (-2)(4 + 5) + (5 - 6)\] \[= -18 - 1\] \[= -19\]

Step 3: Formulate Matrices A_a, A_b, A_c

Create matrices \(A_a\), \(A_b\), and \(A_c\) by replacing columns in \(A\) with \(B\):\**Matrix \(A_a\):** Replace first column with \(B\):\\[A_a = \begin{bmatrix} 2 & 0 & 1 \ 9 & 2 & -1 \ 22 & 5 & 2 \end{bmatrix}\] \**Matrix \(A_b\):** Replace second column with \(B\):\\[A_b = \begin{bmatrix} -2 & 2 & 1 \ 1 & 9 & -1 \ 3 & 22 & 2 \end{bmatrix}\] \**Matrix \(A_c\):** Replace third column with \(B\):\\[A_c = \begin{bmatrix} -2 & 0 & 2 \ 1 & 2 & 9 \ 3 & 5 & 22 \end{bmatrix}\]

Step 4: Calculate Determinant of A_a

The determinant of \(A_a\), \(\text{det}(A_a)\), is calculated as:\\[\text{det}(A_a) = \begin{vmatrix} 2 & 0 & 1 \ 9 & 2 & -1 \ 22 & 5 & 2 \end{vmatrix}\] \[= 2(2 \times 2 + 1 \times 5) - 0(9 \times 2 - 1 \times 22) + 1(9 \times 5 - 2 \times 22)\] \[= 2(4 + 5) + (45 - 44)\] \[= 2 \times 9 + 1\] \[= 18 + 1\] \[= 19\]

Step 5: Calculate Determinant of A_b

The determinant of \(A_b\), \(\text{det}(A_b)\), is calculated as:\\[\text{det}(A_b) = \begin{vmatrix} -2 & 2 & 1 \ 1 & 9 & -1 \ 3 & 22 & 2 \end{vmatrix}\] \[= (-2)(9 \times 2 - (-1)\times 22) - 2(1 \times 2 - (-1) \times 3) + 1(1 \times 22 - 9 \times 3)\] \[= (-2)(18 + 22) - 2(2 + 3) + (22 - 27)\] \[= (-2)(40) - 10 - 5\] \[= -80 - 10 - 5\] \[= -95\]

Step 6: Calculate Determinant of A_c

The determinant of \(A_c\), \(\text{det}(A_c)\), is calculated as:\\[\text{det}(A_c) = \begin{vmatrix} -2 & 0 & 2 \ 1 & 2 & 9 \ 3 & 5 & 22 \end{vmatrix}\] \[= (-2)(2 \times 22 - 9 \times 5) - 0(1 \times 22 - 9 \times 3) + 2(1 \times 5 - 2 \times 3)\] \[= (-2)(44 - 45) + 2(5 - 6)\] \[= (-2)(-1) + 2(-1)\] \[= 2 - 2\] \[= 0\]

Step 7: Compute Solutions Using Cramer's Rule

Using Cramer's rule, we find each variable using the formula \(x_i = \frac{\text{det}(A_i)}{\text{det}(A)}\).\For \(a\): \(a = \frac{\text{det}(A_a)}{\text{det}(A)} = \frac{19}{-19} = -1\).\For \(b\): \(b = \frac{\text{det}(A_b)}{\text{det}(A)} = \frac{-95}{-19} = 5\).\For \(c\): \(c = \frac{\text{det}(A_c)}{\text{det}(A)} = \frac{0}{-19} = 0\).

Key Concepts

Determinants

When engaging with matrix algebra, determinants play a pivotal role. A determinant is a special number that can be calculated from a square matrix. In this exercise, we are working with a 3x3 matrix, which makes the calculation of the determinant slightly more complex than a 2x2 matrix.
The determinant gives us valuable information, such as solving systems of linear equations using Cramer's Rule, which relies on determinants. To find the determinant of a 3x3 matrix, we need to utilize a process known as cofactor expansion. It involves selecting any row or column, but typically the first row is chosen for simplicity.

• Pick a row or column.
• Multiply each element in that row or column by its cofactor.
• Sum these products.

This method allows us to determine whether the system of equations has a unique solution, infinitely many solutions, or no solution at all. A non-zero determinant suggests a unique solution exists—a vital concept in linear algebra.

Systems of Linear Equations

Systems of linear equations are sets of equations with multiple variables that can be solved together. They are extremely common in mathematics and appear frequently in real-world applications like physics, engineering, and economics.
In a system of linear equations, each equation represents a straight line (or a plane in three dimensions), and the solution to the system is the intersection of these lines (or planes). The goal of solving such a system is to find values for the variables that satisfy all equations at once.

• If the system has a single solution, the lines or planes intersect at a single point.
• If it has no solution, the lines are parallel, or the planes do not intersect.
• If there are infinite solutions, the lines or planes coincide.

Cramer's Rule provides a straightforward way of solving such systems if the coefficient matrix is square and the determinant is non-zero.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices—a rectangular array of numbers. Operations like addition, subtraction, multiplication, and finding determinants are fundamental to matrix algebra.
Matrices are an invaluable tool in solving systems of linear equations. They allow for a compact representation of complex equations and can be manipulated in ways that individual equations cannot. For example, rewriting a system of equations in matrix form enables us to use efficient computational methods and apply powerful algebraic tools like Cramer's Rule.

• A matrix can represent multiple linear equations simultaneously.
• The coefficients of the variables are assembled into a matrix which can be analyzed for properties like rank and determinant.
• Solving for variables becomes simplified through matrix operations.

Understanding matrix algebra is crucial for tackling large systems, especially in scientific computing and data analysis.