Problem 11 - Advanced Engineering Mathematics

Question

In Problems 11-14, evaluate the determinant of the given matrix using the result $$ \left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|=5 $$ $$ \mathbf{A}=\left(\begin{array}{lll} a_{3} & a_{2} & a_{1} \\ b_{3} & b_{2} & b_{1} \\ c_{3} & c_{2} & c_{1} \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of matrix \( \mathbf{A} \) is \( -5 \).

Step 1: Define the Determinant Property

The determinant of a matrix with swapped rows or columns has a known property: if two rows (or columns) of a square matrix are swapped, the determinant of the matrix changes sign. So, if you have a matrix \( A \) with determinant \( |A| = D \), then swapping two rows (or two columns) results in \( -D \).

Step 2: Apply the Determinant Property

The given result is the determinant \( \left| \begin{array}{lll} a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \ c_{1} & c_{2} & c_{3} \end{array} \right| = 5 \). The matrix \( \mathbf{A} \) has its first and third columns swapped compared to the original matrix, equivalent to performing two column swaps (first with third, and second with second which is basically no change). Each column swap changes the sign of the determinant, thus two swaps keep it the same sign.

Step 3: Calculate Determinant of Matrix A

Based on Step 2, swapping the columns in \( \mathbf{A} \) does not change the sign twice, it only changed it once as there are only three columns forming two swaps effect. Hence, the determinant of matrix \( \mathbf{A} \) is \( -5 \) (since it swaps only one column against the canonical basis orientation).

Step 4: Conclusion

Therefore, the determinant of the matrix \( \mathbf{A} \) is \( -5 \) by the determinant properties we discussed.

Key Concepts

Determinant Properties

The determinant of a matrix is a critical concept in linear algebra with various properties that greatly influence matrix operations. One key property is that swapping two rows or two columns of a matrix inverts the sign of its determinant. For example, if a matrix \( A \) has a determinant \( |A| = D \), and you swap any two rows or columns, the new determinant becomes \( -D \). This property is beneficial when calculating determinants manually, as reducing a matrix to simpler forms often involves row and column swaps, thus allowing for easier computation. Additionally, if a matrix row or column consists entirely of zeros, its determinant is zero. Such properties streamline the matrix simplification process before manually calculating the determinant, making understanding these rules vital for mathematical efficiency.

It's important to remember that determinants only apply to square matrices, meaning matrices having the same number of rows and columns. If a single row or column of a square matrix is duplicated, the determinant also equals zero. This is because a matrix with two identical rows (or columns) has a reduced rank, failing to span the appropriate vector space dimension.

Matrix Operations

Matrix operations are fundamental to linear algebra. Basic operations include addition, subtraction, and multiplication. When working with these operations, it is crucial to remember the rules that govern matrix dimensions. Matrices can only be added or subtracted if they share the same dimensions, meaning they must have the same number of rows and columns.

Multiplying matrices is slightly more complex. Unlike numbers, the order of multiplication matters. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. This results in a product matrix with dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. For example, multiplying a \( 2 \times 3 \) matrix by a \( 3 \times 4 \) matrix results in a \( 2 \times 4 \) product matrix.

• Transposing a Matrix: This involves flipping a matrix over its diagonal, switching its row and column indices. The transpose of a \( m \times n \) matrix is an \( n \times m \) matrix.
• Calculating Inverses: For a square matrix, you can find an inverse only if its determinant is non-zero. This involves various methods such as row reduction or using the adjugate matrix and the determinant.

Understanding these operations and how they interrelate with determinants can help in solving complex linear algebra problems.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear maps, and their representations using matrices and vector spaces. A central focus is studying vector spaces, which are collections of vectors that can be added together and scalar multiplied.

Fundamental concepts in linear algebra include:

• Vectors and Vector Spaces: Vectors are objects that can be added together or multiplied by scalars to form new vectors. Vector spaces are sets equipped with these operations, obeying specific rules such as distributive and associative properties.
• Linear Transformations: These are functions between vector spaces that respect vector addition and scalar multiplication, often expressed as matrices.
• Eigenvalues and Eigenvectors: These are crucial in understanding linear transformations. An eigenvector of a matrix is a vector that, when multiplied by the matrix, results in a scalar multiple of itself. The corresponding scalar is the eigenvalue.

Linear algebra is pivotal for various applications including computer science, physics, engineering, and more. It provides tools for modeling and solving practical problems like systems of linear equations, handling large data sets, and performing complex calculations necessary in science and engineering fields.