Question

Use the determinant \(\left|\begin{array}{ccc}4 & 0 & -1 \\ 2 & 1 & -7 \\ 3 & 3 & 9\end{array}\right|\) to verify the first four properties of determinants.

Short answer

The determinant is 117, confirming each of the first four properties through specific examples.

Step-by-step solution

Calculate the Determinant

To calculate the determinant of the 3x3 matrix, use the formula for a 3x3 determinant: \[ |A| = a(ei − fh) − b(di − fg) + c(dh − eg) \] Let's denote the matrix as \[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \], so our given matrix is \( \begin{array}{ccc} 4 & 0 & -1 \ 2 & 1 & -7 \ 3 & 3 & 9 \end{array} \). Substituting into the formula, we get: \[ |A| = 4(1 \times 9 - (-7) \times 3) - 0(2 \times 9 - (-7) \times 3) - 1(2 \times 3 - 1 \times 3) \] Calculate each term: \[ |A| = 4(9 + 21) - 0(...) - 1(6 - 3) \] \[ |A| = 4 \times 30 - 3 \] \[ |A| = 120 - 3 \] \[ |A| = 117 \] The determinant of the matrix is 117.

Verify Property 1 (Interchanging Rows)

The first property states that if two rows of a determinant are interchanged, the determinant changes its sign. We'll interchange the first and second rows and recalculate the determinant: \( \begin{array}{ccc} 2 & 1 & -7 \ 4 & 0 & -1 \ 3 & 3 & 9 \end{array} \). The determinant will be: \[ |B| = 2(0 \times 9 - (-1) \times 3) - 1(4 \times 9 - (-1) \times 3) + (-7)(4 \times 3 - 0 \times 3) \] Calculate: \[ |B| = 2 \times 3 - 1 \times 39 - 7 \times 12 \] \[ |B| = 6 - 39 - 84 \] \[ |B| = -117 \] The determinant is -117, which is the negative of the original determinant, verifying the property.

Verify Property 2 (Determinant of Triangular Matrices)

The second property states that the determinant of a triangular matrix (upper or lower) is the product of its diagonal elements. Our original matrix is not triangular. Let's consider a lower triangular matrix as an example: \( \begin{array}{ccc} 4 & 0 & 0 \ 2 & 1 & 0 \ 3 & 3 & 9 \end{array} \). Calculate its determinant: \[ |C| = 4 \times 1 \times 9 = 36 \] The determinant is the product of the diagonal elements: 4, 1, and 9.

Verify Property 3 (Determinant with a Row of Zeroes)

The third property states that if a matrix has a row of zeroes, its determinant is zero. Create a matrix with a row of zeroes: \( \begin{array}{ccc} 4 & 0 & -1 \ 0 & 0 & 0 \ 3 & 3 & 9 \end{array} \). The determinant is: \[ |D| = 0(3 \times 9 - 3 \times (-1)) = 0 \] Since the second row is zero, the determinant is zero, confirming the property.

Verify Property 4 (Scalar Multiplication of a Row)

The fourth property states that if one row of a determinant is multiplied by a scalar, the determinant is also multiplied by that scalar. Multiply the first row by 2: \( \begin{array}{ccc} 8 & 0 & -2 \ 2 & 1 & -7 \ 3 & 3 & 9 \end{array} \). The determinant will be: \[ |E| = 8(1 \times 9 - (-7) \times 3) - 0(...) - 2(2 \times 3 - 1 \times 3) \] Calculate: \[ |E| = 8 \times 30 - 6 \] \[ |E| = 240 - 6 \] \[ |E| = 234 \] The original determinant was 117. After doubling the first row, the determinant is doubled to 234, verifying the property.

Key concepts

Properties of Determinants

Determinants are a fascinating aspect of matrix algebra. Understanding them helps in solving systems of linear equations and in transformations within linear algebra. A few fundamental properties help make calculations easier and verify solutions:

• Interchanging Rows or Columns: When you interchange two rows or columns, the sign of the determinant changes. For example, after swapping two rows, if the original determinant was 117, the new determinant becomes -117.
• Triangular Matrices: For triangular matrices (upper or lower), the determinant is simply the product of the diagonal elements. This property helps in quick determinant calculations.
• Row or Column of Zeroes: If a matrix contains a row or column full of zeroes, the determinant is zero. This can simplify calculations immensely when identifying zero determinants without computation.
• Scalar Multiplication: If a row or column is multiplied by a scalar, the entire determinant is multiplied by that scalar as well. This is useful when scaling matrices for various applications.

These properties are great tools for simplifying and verifying the determinants of matrices.

Matrix Algebra

Matrix algebra involves operations with matrices, such as addition, subtraction, multiplication, and finding the determinant. Determinants, in particular, play a crucial role.

• Addition: While determinants do not directly involve addition, understanding matrix addition helps form larger matrices for determinant calculations when matrices are part of larger systems.
• Multiplication: This is more directly related to determinants. The determinant of a product of two matrices is equal to the product of their determinants, i.e., \( |AB| = |A||B| \), yielding efficient calculations for complex matrix systems.
• Inverse Matrices: A matrix is invertible if its determinant is non-zero. Understanding the determinant helps in finding the inverse and resolving linear systems.

These operations are foundational in solving real-world problems, especially in physics and engineering, where matrices are prevalent.

Linear Algebra

Linear algebra is an area of mathematics dealing with vectors, matrices, and, importantly, linear transformations. Determinants provide insights into these transformations.

• System of Equations: Determinants assist in determining the consistency of linear systems. If the determinant of the coefficient matrix is zero, the system of equations might be inconsistent or have infinite solutions.
• Eigenvalues and Eigenvectors: These concepts are significant in determining characteristics of matrices. The determinant can be pivotal in solving the characteristic equation to find eigenvalues, and subsequently, eigenvectors.
• Transformations: Determinants help in understanding the scale of transformations. A determinant of zero signifies a transformation that collapses the space to a lower dimension (e.g., 3D to 2D).

Understanding these concepts is essential for advanced studies in various fields, including data science and machine learning.

3x3 Matrices

3x3 matrices are commonly used in various fields due to their suitability for representing simple transformations and systems.

• Formula for Determinant: The formula \( |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \) is used for calculating the determinant of a 3x3 matrix. This allows determining important characteristics, like invertibility and consistency in systems.
• Applications: 3x3 matrices are vital in physics to express rotations, in computer graphics for calculating cross products, and in engineering for various transformations.
• Solving Systems: By understanding the determinant, solutions to 3-variable linear equations can be efficiently computed using methods like Cramer's rule.

Mastering 3x3 matrices lays the groundwork for higher-dimensional matrix operations, crucial in modern computational applications.