Problem 31 - Precalculus: Mathematics for Calculus

Question

Use a graphing calculator to find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrrr} 1 & 10 & 2 & 7 \\ 2 & 18 & 18 & 13 \\ -3 & -30 & -4 & -24 \\ 1 & 10 & 2 & 10 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0, so the matrix does not have an inverse.

Step 1: Input the Matrix

Enter the given matrix into the graphing calculator. For this exercise, make sure you correctly format the matrix: \( A = \left[\begin{array}{rrrr} 1 & 10 & 2 & 7 \ 2 & 18 & 18 & 13 \ -3 & -30 & -4 & -24 \ 1 & 10 & 2 & 10 \end{array}\right] \). While inputting, ensure each number is correctly placed in its respective row and column.

Step 2: Find the Determinant

Use the calculator's function for determinant calculation (usually found in the matrix menu). Select matrix \( A \) and calculate the determinant by choosing the appropriate option. For most calculators, this involves selecting the 'det' operation followed by the matrix name. Once computed, observe the result on the display.

Step 3: Interpret the Determinant

Check if the determinant obtained is zero or not. If the determinant is zero, the matrix does not have an inverse. If it is not zero, the matrix does have an inverse (though you do not need to calculate it).

Key Concepts

Graphing Calculator

A graphing calculator is an invaluable tool in mathematics, especially when working with matrices. It simplifies complex calculations such as finding the determinant of a matrix efficiently. Here’s how it helps:

• First, ensure you have entered the matrix correctly. Double-check each value fits in its corresponding row and column.
• Navigate to the calculator’s matrix menu. This is where you'll usually find options for inputting matrices and performing operations.
• Use the built-in function for finding determinants. Most graphing calculators come with a specific button or operation command often labeled as 'det'.

This technology makes it less prone to manual errors when calculating determinants, making your workflow easier and more accurate. Having a graphing calculator allows students to focus on understanding concepts rather than getting bogged down by arithmetic.

Matrix Inverse

The inverse of a matrix is an important concept in linear algebra. However, not every matrix has an inverse. A matrix, in order to have an inverse, must be square (same number of rows and columns) and its determinant must not be zero.

• Here, checking the determinant of a matrix quickly helps determine if it is invertible.
• If the determinant is zero, the matrix lacks an inverse and is considered singular.
• If the determinant is non-zero, the matrix has an inverse, although you may not need to calculate it for this task.

Understanding the concept of the matrix inverse shows the practical application of determinants and how they relate to the ability to solve systems of equations.

Determinant Calculation

Determinants are a foundational concept in matrix operations. They provide key insights into the properties of matrices.

• Calculating the determinant of a matrix involves certain arithmetic based on the values within the matrix.
• For a 4x4 matrix like this one, the calculation uses a combination of its smaller 3x3 submatrices.
• A determinant of zero means the matrix is singular and not invertible, putting a stop to finding its inverse.

In this specific problem, using a graphing calculator instead of manual computation aids accuracy and efficiency, as calculating larger matrices by hand can be prone to error.