Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{aligned} &0.1 x+0.2 y+0.3 z=-1.4\\\ &0.1 x-0.2 y+0.3 z=0.6\\\ &0.4 y+0.9 z=-2 \end{aligned} $$

Step-by-Step Solution

Answer

Solve by finding the inverse matrix and multiplying by the constant vector to find \( x, y, z \).

Step 1: Write the system in matrix form

Let's represent the system of equations as a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \). Here,\[A = \begin{pmatrix} 0.1 & 0.2 & 0.3 \ 0.1 & -0.2 & 0.3 \ 0 & 0.4 & 0.9 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \ y \ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -1.4 \ 0.6 \ -2 \end{pmatrix} \]

Step 2: Find the inverse of matrix A

Calculate the inverse \( A^{-1} \) of the matrix \( A \). First, compute the determinant of \( A \). Then, find the adjoint of \( A \) and divide each element by the determinant. The determinant can be calculated by using the formula for a 3x3 matrix.

Step 3: Verify if A is invertible

Calculate the determinant of \( A \) using the formula \(\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\), where \( a = 0.1 \), \( b = 0.2 \), \( c = 0.3 \), \( d = 0.1 \), \( e = -0.2 \), \( f = 0.3 \), \( g = 0 \), \( h = 0.4 \), and \( i = 0.9 \). If \( \det(A) eq 0 \), then \( A \) is invertible.

Step 4: Compute the determinant of A

\[ \det(A) = 0.1((-0.2)(0.9) - (0.3)(0.4)) - 0.2((0.1)(0.9) - (0.3)(0)) + 0.3((0.1)(0.4) - (-0.2)(0)) \]\[ \det(A) = 0.1(-0.18 - 0.12) - 0.2(0.09) + 0.3(0.04) \]\[ \det(A) = 0.1(-0.3) - 0.018 + 0.012 \]\[ \det(A) = -0.03 - 0.018 + 0.012 = -0.036 \]Thus, the determinant is \( -0.036 \), and \( A \) is invertible.

Step 5: Calculate the inverse of A using adjugate method

Calculate the adjugate of \( A \) and then the inverse \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \). Find each minor, cofactor, and then adjugate by transposing the cofactor matrix. Divide by the determinant calculated previously.

Step 6: Solve for \( \mathbf{x} \) using \( A^{-1} \mathbf{b} \)

Multiply the inverse matrix \( A^{-1} \) by the vector \( \mathbf{b} \) to find \( \mathbf{x} \). This will give you the values for \( x \), \( y \), and \( z \). Carry out the matrix multiplication to complete this.

Key Concepts

Inverse of a Matrix

To solve a system of equations using matrices, we often need to find the inverse of a matrix. The inverse of a matrix, denoted as \( A^{-1} \), is a matrix that, when multiplied by the original matrix \( A \), results in the identity matrix. This property can be crucial for solving linear equations in matrix form.

• First, ensure that your matrix is square. In this case, a 3x3 matrix fits this requirement.
• Next, calculate its determinant. A matrix has an inverse only if its determinant is not zero.
• If the determinant is not zero, calculate the adjugate matrix and divide by the determinant to get the inverse.

Remember, not every matrix has an inverse. If the determinant is zero, the matrix is said to be "singular," and alternative methods for solving the equations must be used. If you find the inverse, place it in a matrix equation \( A^{-1}\mathbf{b} = \mathbf{x} \) to solve for the variables.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. For a 3x3 matrix, the determinant helps in assessing whether the matrix is invertible. A non-zero determinant indicates that a matrix is possibly invertible, while a zero determinant means it is not.
The formula for a 3x3 matrix determinant \( det(A) \) with elements \( a_{11}, a_{12}, ..., a_{33} \) is:\[ det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Each of these terms represents the product of elements and the difference between minor determinants. Breaking it down further can make sure you totally grasp the idea:

• Identify your elements in the matrix. Assign arbitrary names like \( a, b, c, d, e, f, g, h, i \) for each in sequence.
• Plug these into the formula, ensuring careful arithmetic and attention to negative signs.

In our case, the determinant was calculated as \(-0.036\), indicating \( A \) is invertible as it is not zero.

Matrix Equation

A matrix equation is an efficient way to represent and solve linear systems of equations. For the given system, it can be written in the form \( A\mathbf{x} = \mathbf{b} \), where:

• \( A \) is a matrix representing the coefficients of the variables in the equations.
• \( \mathbf{x} \) is the vector of the unknowns \( x, y, z \).
• \( \mathbf{b} \) is the vector representing the solutions on the right-hand side.

To solve the system, you multiply the inverse of matrix \( A \), if it exists, with vector \( \mathbf{b} \). This requires matrix multiplication, a systematic procedure where each element of the resulting vector is a sum of products.
Two matrices are multiplied by taking the dot product of rows and columns. Once \( A^{-1}\mathbf{b} \) is calculated, you obtain \( \mathbf{x} \), giving the solutions for the variables \( x, y, z \) in the system.

3x3 Matrices

A 3x3 matrix is a grid of numbers with 3 rows and 3 columns, often used in linear algebra to solve systems of equations involving three variables. These matrices provide a structured method to solve complex equations.
When engaging with 3x3 matrices, keep these points in mind:

• A 3x3 matrix is particularly useful when dealing with three variables due to its size and form.
• Matrix operations such as finding the inverse, determinant, and multiplication follow specific rules.
• Each position in the matrix is significant, and changes in the elements affect the solution.

The process of solving a three-variable linear system with this matrix involves:

• Creating the matrix from given equations.
• Finding the determinant and maybe its inverse.
• Solving the matrix equation system.

Understanding these foundations helps to handle larger and more complex systems in mathematics. Always approach them step-by-step, carefully managing calculations and matrix operations. Good grasp of these concepts ensures accurate solutions.