Problem 14 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix using the result \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\\ c_{1} & c_{2} & c_{3}\end{array}\right|=5\). $$ \mathbf{D}=\left(\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the determinant of matrix \( \mathbf{D} \) given the determinant of another matrix \( \begin{bmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{bmatrix} \) is 5.

Step 2: Compare Matrices

Compare the given matrix \( \mathbf{D} \) and the matrix with a known determinant. Observe that \( \mathbf{D} \) is obtained from the known matrix by transposition. The original matrix is\[\begin{bmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{bmatrix}\]and \( \mathbf{D} \) is\[\begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}\]which is its transpose.

Step 3: Recall Properties of Determinants

Recall that the determinant of a matrix does not change if the matrix is transposed. In other words, for any square matrix \( A \), \( \det(A) = \det(A^T) \).

Key Concepts

Matrix Transposition

Matrix transposition is an operation where the rows of a matrix become columns and vice versa. When you transpose a matrix, you essentially flip it over its diagonal. For instance, if you have a matrix \big(A\big)\ with rows \left(r_1, r_2, r_3\right)\, \the transposed version, \(A^T\), will have columns \left(c_1, c_2, c_3\right),\ where each \(c_i\) equals \(r_i\).

Let's see this in action using a 3x3 matrix example:

• Original: \left[\begin{array}{ccc}a & b & c \ d & e & f \ g & h & i\end{array}\right]\
• Transpose: \left[\begin{array}{ccc}a & d & g \ b & e & h \ c & f & i\end{array}\right]\

When you transpose the matrix, the position of each element changes, but the set of elements remains the same. Transposition is a fundamental concept widely used in various linear algebra problems due to its simplicity and symmetry.

Properties of Determinants

Determinants play a crucial role in linear algebra, especially in solving systems of linear equations, finding inverses of matrices, and understanding geometrical properties. One of the essential properties of determinants is their behavior during matrix transposition.

A key formula to remember is: \[ \det(A) = \det(A^T) \].
In simple terms, this means that transposing a matrix does not alter its determinant. So, whether you calculate the determinant of a matrix directly, or you first transpose the matrix and then calculate it, you will get the same result.

This property is particularly helpful when comparing two matrices where one is the transpose of the other, as shown in the original exercise. The knowledge that their determinants are identical simplifies calculations and helps in verifying results. This feature of determinants reflects the symmetry and balance inherent in many algebraic structures.

Linear Algebra Concepts

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. It is the foundation for most areas of mathematics and is prevalent in various practical fields such as engineering, physics, computer science, and economics.

One of the key concepts in linear algebra is the matrix, which is a rectangular array of numbers or functions. You can perform various operations on matrices, such as addition, multiplication, and transposition. Understanding matrices is crucial because they represent systems of linear equations, transformations, and more.

Determinants are another fundamental concept. They provide scalar values that help assess certain properties of matrices, such as invertibility and volume scaling in geometric transformations. Determinants are calculated from the elements of a square matrix and give insight into the matrix's characteristics. These insights include whether a system of equations has a unique solution or if vectors are linearly independent.

• Matrices encapsulate data and transformations efficiently.
• Determinants determine uniqueness and solvability in systems of equations.
• Linear algebra bridges algebraic concepts with geometric visualization.

The collective understanding of these concepts offers powerful tools for problem-solving in theoretical and applied mathematics contexts.