Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 0.1 & 0.3 & 0.2 \\ -0.3 & -0.2 & 0.1 \\ 1 & 2 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is approximately -0.18.

Step 1: Understand the Matrix

Our first row in the matrix is 0.1, 0.3, 0.2. Our second row has -0.3, -0.2, 0.1. Our third row has the numbers 1, 2, 3.

Step 2: Implement the Determinant Formula for a Three by Three Matrix

To calculate the determinant of the given matrix, use the formula: \(det(A) = aei + bfg + cdh - ceg - bdi - afh\), where a, b, c, etc., refer to the elements of the matrix according to this layout\[\begin{bmatrix} a&b&c\ d&e&f\ g&h&i\end{bmatrix}\]For our given matrix, this becomes:\[\begin{bmatrix} 0.1&0.3&0.2\ -0.3&-0.2&0.1\ 1&2&3\end{bmatrix}\]So a = 0.1, b = 0.3, c = 0.2, d = -0.3, e = -0.2, f = 0.1, g = 1, h = 2, i = 3.

Step 3: Calculate the determinant

Substitute the values from our matrix into the formula to get the determinant:\(det(A) = 0.1*(-0.2)*3 + 0.3*0.1*1 + 0.2*(-0.3)*2 - 0.2*0.1*1 - 0.3*(-0.2)*2 - 0.1*(-0.3)*3\)

Key Concepts

Determinant Formula

Understanding the determinant of a matrix is key to grasping many concepts in linear algebra and its applications. The determinant is a scalar value that summarizes certain properties of a matrix and can be particularly useful in solving systems of linear equations, analyzing linear transformations, and more.

The determinant of a 2x2 matrix can be computed by taking the product of the diagonals and subtracting one from the other, symbolically given as \(ad - bc\) for matrix \[\begin{array}{cc}a & b \ c & d\end{array}\]. In the case of a 3x3 matrix, which is present in our exercise, the formula becomes more complex.

To calculate the determinant of a 3x3 matrix, we use a method involving cross multiplication of the rows and columns. The formula for such a matrix represented as \[\begin{bmatrix}a & b & c \ d & e & f \ g & h & i\end{bmatrix}\] is given by \(a\cdot e\cdot i + b\cdot f\cdot g + c\cdot d\cdot h - c\cdot e\cdot g - b\cdot d\cdot i - a\cdot f\cdot h\).

Each term in the calculation is a product of diagonal elements, after considering the signs associated with their position. It's crucial to pay careful attention to positive and negative signs during computation to arrive at the correct determinant value.

Matrix Algebra

Matrix algebra is a fundamental part of linear algebra, dealing with various operations that can be performed on matrices such as addition, subtraction, multiplication, and finding inverses.

Key concepts in matrix algebra include:

• Matrix Addition and Subtraction: This is performed element-wise and requires matrices of the same dimensions.
• Matrix Multiplication: This involves the dot product of the rows of the first matrix with the columns of the second matrix.
• Matrix Transposition: This is flipping a matrix over its diagonal, which turns the row index of each element into a column index, and vice versa.
• Inverse of a Matrix: The inverse of a matrix A, denoted as \(A^{-1}\), is a matrix that, when multiplied with A, yields the identity matrix.

Understanding how these operations work with different sizes and types of matrices is crucial. Not every operation is possible for every matrix, for instance, only square matrices can have a determinant or an inverse.

Graphing Utility

Graphing utilities are powerful tools that can assist students and professionals alike in visualizing and solving complex mathematical problems. They're capable of plotting equations, systems of equations, and inequalities on a coordinate plane.

Here's how a graphing utility can be beneficial:

• Analyze the graph of functions and intersecting points.
• Explore transformations and their effects on graphs.
• Calculate numerical values, such as determinants, that may be cumbersome or prone to error by hand.

When it comes to finding the determinant of a matrix, graphing utilities often have built-in functions that allow for quick and accurate computation. This alleviates any potential manual error and serves as a verification method for hand calculations. Remember, while graphing utilities are beneficial, having a fundamental understanding of the underlying concepts is essential for interpreting the results.