Problem 5 - Algebra 2

Question

Use a graphing calculator to evaluate the determinant of each \(3 \times 3\) matrix. $$ \left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 1.

Step 1: Understand the Determinant for a 3x3 Matrix

The determinant of a 3x3 matrix can be found by applying the rule of Sarrus, or by expanding along a row or column. For a diagonal matrix, the determinant is the product of the diagonal elements.

Step 2: Use the Property of Diagonal Matrices

Since the given matrix is a diagonal matrix with 1's on the diagonal and 0's everywhere else, the determinant is simply the product of its diagonal entries. In this case, it's just 1*1*1.

Step 3: Calculate the Determinant

Multiplying the diagonal entries 1*1*1 gives the result of 1, which is the determinant of the matrix.

Key Concepts

Graphing Calculator in Linear Algebra

Graphing calculators are versatile tools that can handle a variety of mathematical computations and graphing tasks. When it comes to linear algebra, these calculators can effortlessly evaluate the determinant of matrices, including those that are 3x3 in size. Although manually computing the determinant is a valuable skill, using a graphing calculator can save time and reduce errors in complex calculations. To use one for finding a determinant, input the matrix elements into the calculator and utilize the built-in matrix function to obtain the result.

Understanding the 3x3 Matrix

A 3x3 matrix is a square arrangement of nine numbers (or elements) in three rows and three columns. This type of matrix is common in linear algebra and can represent systems of equations, transformations, and more. The arrangement of the numbers within the matrix is crucial since the position of each element affects its properties and the operations that can be performed, such as calculating the determinant, which provides information about the matrix's invertibility and the volume of the geometric transformation it represents.

The Rule of Sarrus

The rule of Sarrus is a simple and clever shortcut to calculate the determinant of a 3x3 matrix. It involves creating two additional copies of the first two columns of the matrix to the right, forming an extended array. Then, the sum of the products of the downward diagonals is subtracted by the sum of the products of the upward diagonals. The result is the determinant of the original matrix. This method is fast and efficient for manual calculations but should not be confused with a general determinant formula for larger matrices—it's specific to 3x3 matrices only.

Diagonal Matrix and Its Determinant

A diagonal matrix is one where all the off-diagonal elements are zero—meaning that all the entries outside the main diagonal from the top left to the bottom right are zeros. The determinant of a diagonal matrix is incredibly straightforward to calculate; it is simply the product of all the diagonal elements. For identity matrices, which are diagonal matrices with ones on the main diagonal, the determinant is always one. This property makes the computation of the determinant significantly faster and highlights the elegance inherent in some of the more specialized matrix forms.

Matrix Determinant Properties

The determinant is a scalar attribute of a square matrix that reveals much about its nature. There are several key properties of determinants. For example, the determinant of a product of matrices equals the product of their determinants, and swapping two rows (or columns) of a matrix will change the sign of its determinant. If a matrix has a row or column of zeros, its determinant is zero, indicating that the matrix does not have an inverse. These properties are valuable in many areas of mathematics, including solving linear equations and understanding geometric transformations.