Problem 38 - Advanced Engineering Mathematics

Question

Suppose \(\mathbf{A}\) is a \(5 \times 5\) matrix for which \(\operatorname{det} \mathbf{A}=-7\). What is the value of \(\operatorname{det}(2 \mathbf{A}) ?\)

Step-by-Step Solution

Answer

-224

Step 1: Understand the Determinant of a Scalar Multiple

When a matrix \( \mathbf{A} \) is multiplied by a scalar, say \( c \), the determinant of the new matrix \( c \mathbf{A} \) can be found by multiplying the determinant of \( \mathbf{A} \) by \( c^n \), where \( n \) is the number of rows (or columns) in the square matrix. In this exercise, \( n = 5 \).

Step 2: Apply the Rule for Scalar Multiplication

Given that \( \operatorname{det}(\mathbf{A}) = -7 \) and \( n = 5 \), we find \( \operatorname{det}(2 \mathbf{A}) \) using the formula: \[ \operatorname{det}(2 \mathbf{A}) = 2^5 \times \operatorname{det}(\mathbf{A}) = 32 \times (-7) \].

Step 3: Perform the Calculation

Calculate \( 32 \times (-7) \) to find the determinant of \( 2 \mathbf{A} \): \[ 32 \times (-7) = -224 \].

Step 4: State the Final Answer

The value of \( \operatorname{det}(2 \mathbf{A}) \) is \( -224 \).

Key Concepts

Scalar Multiplication of Matrices

When you multiply a matrix by a scalar, each element of the matrix is multiplied by that scalar value. This operation is called scalar multiplication of matrices. Take a matrix, say \( \mathbf{A} \), composed of rows and columns. Let's say you multiply it by a scalar \( c \). The result is another matrix where each element is the product of the original element and \( c \).
For a simple illustration, consider matrix \( \mathbf{B} = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \). If we scale \( \mathbf{B} \) by \( 3 \), we get \( 3\mathbf{B} = \begin{bmatrix} 3 \times 1 & 3 \times 2 \ 3 \times 3 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} \).
Scalar multiplication is straightforward and offers a basis for matrix transformations. Understanding this concept is crucial for delving into more complex operations in matrix algebra.

Matrix Algebra

Matrix algebra involves operations like addition, subtraction, multiplication, and finding determinants or inverses of matrices. These operations form the foundation for more advanced topics in linear algebra.

• **Addition and Subtraction**: Requires matrices to have the same dimensions. Element-wise operations are performed.
• **Multiplication**: More complex. Requires number of columns in first matrix to match number of rows in second matrix. Not commutative, meaning \( \mathbf{A}\mathbf{B} eq \mathbf{B}\mathbf{A} \).
• **Determinant**: A scalar value reflecting certain properties of a square matrix. Essential for solving systems of linear equations.

Matrix algebra simplifies complex systems and solves various real-world problems, making it invaluable in science and engineering.

Properties of Determinants

The determinant of a matrix tells us about the scaling factor of the transformation that the matrix represents. It helps to understand several key properties:

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• The determinant helps in finding the eigenvalues of a matrix.
• Determinants are essential in understanding matrix invertibility and linear independence of matrix rows and columns.
• Importantly, for a matrix \( \mathbf{A} \), if \( \operatorname{det}(\mathbf{A}) \) is given, the determinant of a scalar multiple, \( c\mathbf{A} \), becomes \( c^n \times \operatorname{det}(\mathbf{A}) \), where \( n \) is the matrix size.

These properties underscore the importance of determinants in linear transformations and systems of equations. Understanding these can greatly enhance the grasp on linear algebra concepts.