Problem 17 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -1.

Step 1: Identify the matrix

The given matrix is a 3x3 matrix:\[ A = \begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} \]

Step 2: Use the formula for 3x3 determinant

The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is calculated by:\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 3: Substitute the matrix values

For our matrix:\[ a = 1, b = 0, c = 1, d = 0, e = 1, f = 0, g = 1, h = 0, i = 0 \]Substitute these values into the determinant formula:

Step 4: Calculate determinants of minors

Calculate each part of the formula:- \( ei - fh = 1 \times 0 - 0 \times 0 = 0 \)- \( di - fg = 0 \times 0 - 0 \times 1 = 0 \)- \( dh - eg = 0 \times 0 - 1 \times 1 = -1 \)

Step 5: Assemble the determinant

Substitute these results back into the formula:\[ \det(A) = 1 \times 0 - 0 \times 0 + 1 \times (-1) = 0 + 0 - 1 = -1 \]

Step 6: Conclusion

The determinant of the given matrix is \(-1\).

Key Concepts

3x3 Matrix

When dealing with matrices, especially in the context of determinant calculations, understanding what a 3x3 matrix is becomes essential. A matrix is essentially a rectangular array of numbers arranged in rows and columns. In the case of a 3x3 matrix, it consists of 3 rows and 3 columns, resulting in nine elements in total. Taking our example matrix:

• First row: [1, 0, 1]
• Second row: [0, 1, 0]
• Third row: [1, 0, 0]

This layout of numbers allows us to perform specific operations essential in linear algebra, such as calculating determinants or exploring matrix properties like rank and trace. A 3x3 matrix is often denoted as \( A \), and the elements within are referred to by their respective positions, like \( a_{11} \) for the element in the first row and first column.

Determinant Calculation

Calculating the determinant of a matrix is a fundamental operation in matrix algebra that serves various purposes such as solving systems of linear equations and understanding certain properties of the matrix. For a 3x3 matrix like our sample, the determinant provides a scalar value that can give us insights into the matrix’s characteristics, such as whether it is invertible. The specific formula used for a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]This formula is structured around the multiplication of diagonal and cross elements, subtracting their product, allowing for a precise calculation of the determinant. When substituted with actual numbers from a given matrix, it delivers a specific numeric determinant value, as seen in our example which results in \(-1\).

Matrix Algebra

Matrix algebra is an extensive topic dealing with the manipulation and application of matrices. It encompasses various operations such as addition, multiplication, and the calculation of determinants and inverses. Each operation has unique rules that enable us to solve complex mathematical problems efficiently. Within matrix algebra, determinants play a critical role. They allow us to determine the invertibility of a matrix. For example, a matrix with a zero determinant is singular, meaning it doesn't have an inverse. In applications, determinants assist in understanding transformations, finding eigenvalues, and solving linear systems. Algebraic operations using matrices follow specific rules related to their dimensions and entries. Being proficient with matrix algebra, including the understanding and calculation of determinants, is vital for anyone dealing with higher-level mathematics or physics.