Problem 10 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. Suppose \(\mathbf{A}\) is a \(3 \times 3\) matrix such that \(\operatorname{det} \mathbf{A}=2 .\) If \(\mathbf{B}=10 \mathbf{A}\) and \(\mathbf{C}=-\mathbf{B}^{-1}\), then \(\operatorname{det} \mathbf{C}=\) _________.

Step-by-Step Solution

Answer

\(-\frac{1}{2000}\)

Step 1: Understand Matrix B

The matrix \( \mathbf{B} \) is defined as \( 10 \mathbf{A} \). The determinant of \( \mathbf{B} \) can be calculated using the property \( \operatorname{det}(k\mathbf{A}) = k^n \cdot \operatorname{det}(\mathbf{A}) \), where \( n \) is the order of the square matrix \( \mathbf{A} \). Since \( \mathbf{A} \) is a \( 3 \times 3 \) matrix, \( n = 3 \). Therefore, \( \operatorname{det}(\mathbf{B}) = 10^3 \cdot 2 \).

Step 2: Calculate Determinant of B

Calculate the determinant of \( \mathbf{B} \): \[ \operatorname{det}(\mathbf{B}) = 10^3 \times 2 = 1000 \times 2 = 2000 \].

Step 3: Understand Matrix C

Matrix \( \mathbf{C} \) is defined as \( -\mathbf{B}^{-1} \). To find \( \operatorname{det}(\mathbf{C}) \), we use the property \( \operatorname{det}(\mathbf{B}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{B})} \). Also, the determinant of a scalar multiple of a matrix, \( -1 \cdot \mathbf{B}^{-1} \), will be \( -1 \cdot \operatorname{det}(\mathbf{B}^{-1}) \).

Step 4: Calculate Determinant of C

Calculate \( \operatorname{det}(\mathbf{C}) \): \( \operatorname{det}(\mathbf{C}) = -1 \times \frac{1}{\operatorname{det}(\mathbf{B})} = -1 \times \frac{1}{2000} = -\frac{1}{2000} \).

Key Concepts

3x3 Matrix

A **3x3 Matrix** is a square matrix with three rows and three columns. Matrices are commonly used to represent systems of equations and transformations in vectors.
Let's discuss some features of a 3x3 matrix,

• The number of rows and columns is equal, which means it is a square matrix.
• This type of matrix can have a determinant, an important scalar value.

The determinant for a 3x3 matrix has a significant role. It helps in finding out if a matrix is invertible.
For a matrix \[\begin{bmatrix}a & b & c \d & e & f \g & h & i \\end{bmatrix}\]the determinant can be found using the formula: \( ext{det} = a(ei − fh) − b(di − fg) + c(dh − eg)\).
This value helps to understand the scale of a transformation or solve equations if the matrix represents one.

Matrix Properties

**Matrix Properties** offer important insights into how matrices behave and interact with each other. These properties simplify matrix operations and are essential for fields such as computer graphics and physics.

• Determinant of a Matrix: It provides information on the matrix's invertibility and, for a 3x3 matrix, it shows how volume or area scales under the matrix transformation.
• Scalar Multiplication: Scalar multiplication affects the determinant such that \( \text{det}(k\mathbf{A}) = k^n \times \text{det}(\mathbf{A}) \), where \( n \) is the size of the square matrix.
• Inverse of a Matrix: The inverse is denoted \( \mathbf{A}^{-1} \). A matrix must have a non-zero determinant to possess an inverse.

When you multiply a matrix by a scalar, it multiplies each element of the matrix by that scalar, leading to a consistent change across the entire matrix. This property is useful when solving problems involving scaled transformations.

Inverse Matrices

The concept of **Inverse Matrices** is crucial in solving linear equations and finding solutions that involve transformations.
An inverse matrix works like a "reverse" operation.

• Existence: A matrix only has an inverse if its determinant is non-zero.
• Identity Matrix: When a matrix is multiplied by its inverse, the result is an identity matrix, behaving like the number 1 in scalar multiplication.
• Computation: For a 3x3 matrix, the inverse is computed using the matrix's determinant and cofactors.

To find an inverse of a matrix **B** where **C** is -**B**\(^{-1}\), it involves finding the reciprocal of the determinant. Understanding these operations ensures accurate manipulation and transformation of data within matrices.