Problem 13 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Matrix Transformations

Observe that the matrix \( \mathbf{C} \) is derived from the original matrix \( \begin{bmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{bmatrix} \) by applying transformations to its rows. The first row is multiplied by \(-1\) and subtracted from the third row in each element.

Step 2: Use Properties of Determinants

When rows of a matrix are manipulated, such as by multiplying by a constant, certain properties of the determinant apply. Multiply all elements of the first row by \(-1\) results in multiplying the determinant by \(-1\). So, the determinant now becomes \(-|A|\), where \(|A| = 5\), thus: \(-5\).

Step 3: Adjust for Additional Row Operation

In matrix \( \mathbf{C} \), the third row is obtained by subtracting the first row from the third row. Row operations involving adding a multiple of one row to another do not change the determinant's value (based on linear algebra properties). Thus, these operations do not alter the determinant further beyond what has been done in Step 2.

Step 4: Conclude the Determinant of the Transformed Matrix

The determinant of matrix \( \mathbf{C} \) remains \(-5\) after accounting for the transformation operations applied to the rows of original matrix.

Key Concepts

Matrix Transformations

Understanding matrix transformations is essential when working with determinants. In our matrix example, we observe changes involving entire rows. These changes include multiplying a row by a constant or adding rows together, all occurring in a linear algebra context.
Think of each row of a matrix as a vector, and imagine these transformations as operations that change these vectors. When you multiply an entire row by -1, you're essentially reflecting that vector across the origin in a 3D space. This type of transformation directly influences the determinant of the matrix. Any operation on a row can be viewed as manipulating these vector spaces, affecting not only the matrix's structure but its determinant as well.
Remember, with matrix transformations, the effects on the determinant depend on the nature and sequence of operations performed on rows.

Determinant Properties

The properties of determinants are crucial when analyzing any matrix problem. These properties help simplify calculations and understand transformations' effects on determinants.

• Scalar Multiplication: When each element of a row is multiplied by a scalar (such as multiplying by -1 in our example), the determinant of the matrix also gets multiplied by that same scalar.
• Row Addition: Adding or subtracting one row from another doesn't change the value of the determinant. This is essential in simplifying matrix operations without altering its fundamental characteristics.
• Row Operations: Simple row swaps change the sign of the determinant, whereas multiplying by a scalar or adding rows maintains its absolute value after adjustments.

Understanding these core properties allows for predicting how matrix alterations will affect the overall determinant.

Linear Algebra

Linear algebra provides the framework necessary to understand transformations and determinant behaviors in matrices. Within this mathematical branch, matrix operations form the basis of solving many real-world problems.
In our specific matrix, linear algebra concepts help us apply determinant properties effectively, showing how transformations impact the matrix's determinant. The subject introduces tools and techniques like row operations, scalar multiplication, and vector spaces, essential in navigating determinant concepts and matrix manipulations.
Know that in linear algebra, the determinant offers insights into a matrix's invertibility and whether it can be transformed into specific forms, playing a crucial role in understanding broader implications of transformations and vcalculation methods.

Row Operations

Row operations are fundamental in determining how matrices behave, especially when calculating determinants. They include:

• Row Multiplication: Multiplying all elements in a row by a scalar affects the determinant by that factor. For instance, using -1 changes the original determinant's sign.
• Row Addition/Subtraction: Adding a multiple of one row to another doesn't alter the determinant. This property simplifies complex matrix problems while maintaining determinant constancy.
• Row Swapping: Interchanging two rows results in a sign change in the determinant. While powerful, it should be used strategically to maintain the problem's context.

Understanding these operations is vital, as they showcase the flexibility and rules governing matrix transformations. By mastering these techniques, students can efficiently manipulate matrices to find determinants and explore deeper matrix-related concepts.