Problem 13 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Exercise

We need to find the determinant of matrix \( \mathbf{C} \) given the determinant of another matrix \( \mathbf{A} \) \( \left|\begin{array}{lll}a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3\end{array}\right|=5 \). Matrix \( \mathbf{C} \) has transformations applied which may affect the determinant.

Step 2: Identify Matrix Transformations

Matrix \( \mathbf{C} \) is derived from matrix \( \mathbf{A} \) by two row transformations: 1) Row 1 is multiplied by \(-1\), and 2) A linear combination of Row 1 and Row 3 is used to form a new Row 3: \( c_i - a_i \). Each transformation affects the determinant.

Step 3: Calculate the Effect of Row 1 Transformation

Multiplying a row by \(-1\) changes the determinant's sign. So, if we transform Row 1 by multiplying it by \(-1\), the determinant of the new matrix becomes \(-\left(\begin{array}{lll}a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3\end{array}\right)\).

Step 4: Analyze Row 3 Transformation

Row 3 in \( \mathbf{C} \) does not change the determinant because adding a multiple of another row (here, subtracting Row 1 from Row 3) does not alter the determinant's value.

Step 5: Combine Effects on the Determinant

The transformation from the original matrix to \( \mathbf{C} \) results in changing the sign of the determinant, but the value does not change otherwise. Therefore, the determinant of \( \mathbf{C} \) is \(-5 \).

Key Concepts

Matrix Transformations

Matrix transformations are key operations that alter the form of a matrix, often for the purpose of solving equations or analyzing matrix properties. A transformation can include operations like row replacement, row swapping, or scaling rows by a constant. These operations are essential tools in linear algebra, facilitating easier computation of matrix properties like determinants.
In the context of determinants, each type of transformation affects the determinant in specific ways. For example:

• Multiplying an entire row by a non-zero constant changes the determinant by that constant's factor.
• Swapping two rows reverses the sign of the determinant.
• Adding a multiple of one row to another does not change the determinant at all.

Understanding these effects is crucial for effectively applying transformations to solve matrix-related problems. In our exercise, transformations were applied to evaluate how they affect the determinant of a specially formed matrix.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, linear transformations, and systems of linear equations. It provides the theoretical foundation for working with matrices and determinants.
Determinants, a core concept in linear algebra, provide a scalar value that encapsulates several properties of a matrix. For square matrices, the determinant can indicate if the matrix is invertible or not. If the determinant is zero, the matrix does not have an inverse, indicating linear dependency between its rows or columns.
Determinants are also crucial for understanding the volume change of geometric transformations represented by matrices. This application is vital in fields ranging from computer graphics to physics. By grasping linear algebra's concepts and operations within it, students can solve complex systems and understand real-world transformations efficiently.

Row Operations

Row operations are fundamental in manipulating a matrix to solve equations or simplify it for analysis. These operations are efficient tools in transforming matrices for better computational handling.
In the context of determinant evaluation, row operations must be employed with an understanding of their effects. The three main row operations are:

• Row Swapping: This operation swaps two rows and changes the sign of the determinant.
• Row Multiplication: Multiplying a row by a constant scales the determinant by that constant.
• Row Addition: Adding a multiple of one row to another does not affect the determinant's value.

The strategic use of these operations can simplify the process of finding determinants, especially in complex matrices like the exercise provided. Recognizing how each operation affects the determinant aids in quick and accurate calculations, which are important skills in linear algebra.