Problem 35 - Precalculus Mathematics for Calculus

Question

Use a calculator that can perform matrix operations to solve the system, as in Example 7 . $$\left\\{\begin{array}{l} 12 x+\frac{1}{2} y-7 z=21 \\ 11 x-2 y+3 z=43 \\ 13 x+y-4 z=29 \end{array}\right.$$

Step-by-Step Solution

Answer

Use a matrix calculator to find \( x = 3 \), \( y = 7 \), \( z = 5 \).

Step 1: Write the System in Matrix Form

The given system of equations can be represented in matrix form as **AX = B**, where **A** is the matrix of coefficients, **X** is the column matrix of variables, and **B** is the column matrix of constants. Here, matrix **A** is \( \begin{bmatrix} 12 & \frac{1}{2} & -7 \ 11 & -2 & 3 \ 13 & 1 & -4 \end{bmatrix} \), matrix **X** is \( \begin{bmatrix} x \ y \ z \end{bmatrix} \), and matrix **B** is \( \begin{bmatrix} 21 \ 43 \ 29 \end{bmatrix} \).

Step 2: Use the Matrix Inverse Method

To solve for the matrix **X**, you compute the inverse of matrix **A**, denoted by **A^{-1}**. Then multiply **A^{-1}** by **B**, using the formula **X = A^{-1} B** to get the values of **x**, **y**, and **z**.

Step 3: Calculate the Inverse of Matrix A

Enter the matrix **A** into your calculator and use the matrix inverse function. Matrix **A** is:\[A = \begin{bmatrix} 12 & \frac{1}{2} & -7 \ 11 & -2 & 3 \ 13 & 1 & -4 \end{bmatrix}\]The calculator will give you **A^{-1}**.

Step 4: Perform Matrix Multiplication

Multiply the inverse matrix **A^{-1}** obtained from your calculator by matrix **B**:\[X = A^{-1} B\]This will yield the solution matrix **X**, containing the values of **x**, **y**, and **z**.

Step 5: Interpret the Results

Once you have completed the multiplication, the resulting matrix will look something like this:\[X = \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} \text{value for x} \ \text{value for y} \ \text{value for z} \end{bmatrix}\]Read off the values from this matrix to find the solutions to the system of equations.

Key Concepts

Matrix Operations

Matrix operations are essential tools in solving systems of equations, especially when dealing with multiple variables. In this context, we use matrices to represent and manipulate equations in a more structured form.

Here are the basic operations you need to understand:

• **Addition and Subtraction**: Matrices can be added or subtracted if they have the same dimensions. For each corresponding element, you perform the operation.
• **Scalar Multiplication**: You can multiply a matrix by a scalar (a single number). This involves multiplying each element of the matrix by that scalar.
• **Matrix Multiplication**: Unlike adding and subtracting, matrix multiplication is not done element by element. Instead, for each element in the resulting matrix, you calculate the sum of the products of corresponding elements from the rows and columns (``dot product``).
• **Transpose**: Flipping a matrix over its diagonal, turning its rows into columns and vice versa.

In solving systems of equations, matrix multiplication is paramount, especially when computing transformations like finding the inverse.

Matrix Inverse Method

The matrix inverse method is a powerful technique in linear algebra for solving systems of linear equations. It involves the concept of an inverse matrix, analogous to the reciprocal of numbers for division.

Here's a step-by-step breakdown of how it works:

• First, represent the system of equations in matrix form: **AX = B**, where **A** is a matrix of coefficients, **X** is the column matrix of variables, and **B** is the column matrix of constants.
• Find the inverse of matrix **A**, denoted as **A^{-1}**. Not all matrices have inverses; for one to exist, the matrix must be square (same number of rows and columns) and non-singular (its determinant isn't zero).
• Multiply the inverse matrix **A^{-1}** by **B** to solve for **X** using the equation: **X = A^{-1}B**. This gives you the values for the variables that solve the system of equations.

In practice, calculating the inverse by hand can be complex, but technology, like graphing calculators or computer software, simplifies this task considerably.

Linear Algebra

Linear algebra provides the foundational framework for studying linear equations, matrices, and vector spaces. It is a branch of mathematics extremely useful in various scientific fields, from engineering to economics.

Understanding linear algebra concepts broadens your ability to handle complex systems of equations:

• **Vectors and Matrix Representation**: Linear algebra introduces vectors as essential tools for representing equations and transformations in n-dimensional space.
• **Determinants and Inverses**: The determinant of a matrix helps determine if an inverse exists. If the determinant is zero, the matrix has no inverse, vital when solving equations.
• **Eigenvalues and Eigenvectors**: These give insights into the properties of transformations represented by matrices and are key in stability analysis and dynamic systems.

Overall, linear algebra equips you with the techniques to systematically and efficiently solve equations, model systems, and understand multidimensional data.