Problem 12 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix using the result \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\\ c_{1} & c_{2} & c_{3}\end{array}\right|=5\). $$ \mathbf{B}=\left(\begin{array}{rrr} 2 a_{1} & a_{2} & a_{3} \\ 6 b_{1} & 3 b_{2} & 3 b_{3} \\ 2 c_{1} & c_{2} & c_{3} \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of matrix \( \mathbf{B} \) is 30.

Step 1: Understand the matrix transformation

Given that the determinant of the matrix \( A = \left| \begin{array}{ccc} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{array} \right| = 5 \), we have matrix \( \mathbf{B} \) transformed with specific row operations. We need to consider how these transformations affect the determinant.

Step 2: Identify transformations

Matrix \( \mathbf{B} \) is derived from \( A \) by transformations on its rows. Specifically, the first row of \( \mathbf{B} \) is obtained by multiplying the first row of \( A \) by 2. The second row of \( \mathbf{B} \) is achieved by multiplying the second row of \( A \) by 3. The third row of \( \mathbf{B} \) is the same as the third row of \( A \).

Step 3: Apply determinant properties

The determinant's value changes when a row of a matrix is multiplied by a constant. Specifically, multiplying a row by a constant \( k \) results in the determinant being multiplied by \( k \). Thus, multiplying the first row by 2 and the second row by 3 results in \( 2 \times 3 = 6 \) times the original determinant \( \det(A) \).

Step 4: Calculate the determinant of B

Given that \( \det(A) = 5 \), the determinant of \( \mathbf{B} \) is \( 2 \times 3 \times \det(A) = 6 \times 5 = 30 \).

Step 5: Conclusion

The determinant of the matrix \( \mathbf{B} \) is 30.

Key Concepts

Matrix Transformations

Matrices are powerful tools in mathematics that allow us to represent transformations, such as rotations and scaling, in a convenient algebraic format. In linear algebra, a matrix can transform a vector by altering its length or direction, which is especially useful in numerous applications from computer graphics to solving systems of equations. A key aspect of matrix transformations is how they affect the determinant, a special number that tells us important properties about the transformation.

• Scaling: When a matrix row is multiplied by a scalar, it scales the transformation accordingly.
• Reflection and Rotation: Such transformations can also affect orientation or angles in a system.
• Determinants as Volume Interpretation: In geometrical terms, for a 2D or 3D matrix, the determinant can represent the area or volume expansion factor of the transformation.

In essence, understanding these transformations connects algebraic manipulations with geometric interpretations, making it a fascinating study in linear algebra.

Row Operations

In the context of matrices, row operations are fundamental tools that allow us to manipulate matrices without altering the solutions of the associated linear equations. There are three primary types of row operations:

• Row Swapping: Interchanging two rows of a matrix will result in a sign change of the determinant.
• Row Multiplication: Multiplying a row by a non-zero scalar will multiply the determinant by that scalar.
• Row Addition: Adding a multiple of one row to another does not change the determinant.

In the given problem, these concepts become crucial. The determinant of matrix \( A \) is already known. To compute the determinant of matrix \( B \), we recognize that the first and second rows of A were each multiplied by constants, leading to an easy calculation using row multiplication properties.
By: 1. Multiplying the determinant by 2 for the first row, and 2. Multiplying by 3 for the second row,we see how matrix \( B \) takes its determinant of 30 from these operations.

Linear Algebra

Linear Algebra is a branch of mathematics that studies systems of linear equations and their representations through matrices and vectors. It provides the foundation for many mathematical concepts and real-world applications, including physics, engineering, computer science, and data analysis.
At the heart of linear algebra is the concept of vectors and matrices. Vectors are mathematical objects that have both a magnitude and a direction, while matrices are arrays of numbers that represent linear transformations between vector spaces. Determinants are crucial as they indicate whether a matrix is invertible or the nature of solutions for the associated system of equations.
In practice, Linear Algebra helps us:

• Analyze network flows by representing and solving systems of equations.
• Pursue machine learning models, where matrices help manipulate and transform data.
• Optimize engineering problems by transforming and rotating coordinate systems.

Grasping these fundamentals enriches our understanding of mathematical problems and enhances our ability to engage with complex systems analytically.