Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$\left[\begin{array}{rrr} 1.3 & 0.2 & 3.2 \\ 0.2 & 6.2 & 0.2 \\ -0.4 & 4.4 & 0.3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the given matrix obtained from the graphing utility. If it's 0, then the matrix is not invertible.

Step 1: Understand the determinant of a 3x3 matrix

If \(A\) is a 3x3 matrix, its determinant, denoted as \(|A|\), is calculated as \[ |A| = a(ei−fh)−b(di−fg)+c(dh−eg) \] where \(A = \left[\begin{array}{rrr} a & b & c \ d & e & f \ g & h & i \end{array}\right]\).

Step 2: Replace the variables with the values of the given matrix

Replace \(a, b,\) ... \(i\) with the values of the given matrix. In this case, \(a=1.3, b=0.2, c=3.2, d=0.2, e=6.2, f=0.2, g=-0.4, h=4.4,\) and \(i=0.3\).

Step 3: Calculate the determinant using the graphing utility

Now input the expression obtained in step 2 into the graphing utility and obtain the determinant.

Step 4: Interpreting the output

The determinant value outputted by the graphing utility is the determinant of the given matrix. If it is 0, it means that the matrix is not invertible.

Key Concepts

3x3 matrix

A 3x3 matrix is essentially a rectangular array consisting of 9 numbers, arranged in three rows and three columns. It is an example of a square matrix, which simply means the number of rows is the same as the number of columns. Understanding the structure of a 3x3 matrix is key because many mathematical procedures, such as finding the determinant in this exercise, depend on the matrix's dimensions.
A 3x3 matrix can be represented as follows:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i \\end{bmatrix}\]In this matrix:

• Each element, such as \(a, b, c,\) represents a number in the matrix.
• There are three rows: the first row \([a, b, c]\), the second row \([d, e, f]\), and the third row \([g, h, i]\).
• Three columns are also present: first column is \([a, d, g]\), second column \([b, e, h]\), and the third column \([c, f, i]\).

These matrices are commonly used in systems of equations, transformations, and various other applications in mathematics and physics.

matrix operations

Matrix operations encompass a variety of actions that can be performed on matrices to solve mathematical problems or simplify expressions. In the context of a 3x3 matrix, common operations include addition, subtraction, multiplication, and finding the determinant.

• Addition and Subtraction: To add or subtract two matrices, they must be of the same size. We add or subtract corresponding elements to create a new matrix.
• Multiplication: This involves multiplying the rows of the first matrix by the columns of the second matrix and summing the products. This operation can change the dimensions of the resulting matrix.
• Determinant: The determinant is a special number that can be calculated from a square matrix. It is crucial in contexts like system equations and inverse matrices, telling us if a matrix is invertible.

In this exercise, we're particularly interested in finding the determinant of a given 3x3 matrix using matrix operations.

graphing utility

A graphing utility is a handy computational tool designed to enhance mathematical exploration and analysis. It usually comes in the form of a handheld calculator or computer software.
Graphing utilities are not only essential for visualizing mathematical models, but they also simplify complex calculations, like determining the determinant of a matrix.
To find the determinant using a graphing utility:

• First, input the matrix by entering each element as it corresponds to the 3x3 grid structure.
• Use the built-in feature for calculating determinants, often found in the matrix menu.
• Execute the command, and the graphing utility will provide the determinant value.

This automation is beneficial for quickly verifying hand calculations or tackling matrices without manually complex arithmetic.