Problem 8 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. $$ \text { If } \operatorname{det} \mathbf{A}=6 \text { and } \operatorname{det} \mathbf{B}=2 \text {, then } \operatorname{det} \mathbf{A} \mathbf{B}^{-1}= $$ _________

Step-by-Step Solution

Answer

3

Step 1: Understand the determinant property for the product of matrices

When determining the determinant of the product of two matrices, the following relation applies: \( \operatorname{det}(\mathbf{A}\mathbf{B}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B}) \). However, in this problem, we have \( \mathbf{B}^{-1} \) instead of \( \mathbf{B} \).

Step 2: Apply the property of the determinant of an inverse matrix

It is a known property that the determinant of the inverse of a matrix \( \mathbf{B} \) is the reciprocal of the determinant of \( \mathbf{B} \). That is, \( \operatorname{det}(\mathbf{B}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{B})} \). Given that \( \operatorname{det}(\mathbf{B}) = 2 \), it follows that \( \operatorname{det}(\mathbf{B}^{-1}) = \frac{1}{2} \).

Step 3: Combine the properties

Now combine the determinant properties. According to the rule for the product of matrices: \( \operatorname{det}(\mathbf{A}\mathbf{B}^{-1}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B}^{-1}) \). Substituting the known values: \( \operatorname{det}(\mathbf{A}) = 6 \) and \( \operatorname{det}(\mathbf{B}^{-1}) = \frac{1}{2} \).

Step 4: Calculate the determinant

Perform the multiplication: \( \operatorname{det}(\mathbf{A}\mathbf{B}^{-1}) = 6 \cdot \frac{1}{2} = 3 \). Therefore, the determinant of \( \mathbf{A} \mathbf{B}^{-1} \) is \( 3 \).

Key Concepts

Inverse Matrices

Inverse matrices are an essential concept in linear algebra, often appearing in contexts like solving systems of equations and transformations. The inverse of a matrix \( \mathbf{B} \), denoted as \( \mathbf{B}^{-1} \), is a matrix that, when multiplied by \( \mathbf{B} \), yields the identity matrix: \( \mathbf{B} \mathbf{B}^{-1} = \mathbf{I} \).

• The identity matrix \( \mathbf{I} \) acts as the "1" in matrix operations, leaving any matrix unchanged when multiplied by it.
• Not all matrices have inverses. A matrix must be square and have a non-zero determinant to be invertible.

The determinant of an inverse matrix can be calculated as the reciprocal of the determinant of the original matrix. If \( \operatorname{det}(\mathbf{B}) = 2 \), then \( \operatorname{det}(\mathbf{B}^{-1}) = \frac{1}{2} \). This property is crucial in computations involving inverse matrices.

Determinant Properties

Determinants have several important properties that make them valuable in matrix calculations. They give insights into the matrix's characteristics, such as invertibility and volume distortion effects during transformations. When dealing with matrix multiplication, the determinant of the product of two matrices is the product of their determinants: \( \operatorname{det}(\mathbf{A} \mathbf{B}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B}) \). This property allows for straightforward computation of complex matrix expressions. Additionally, if a matrix's determinant is zero, the matrix is singular, meaning it has no inverse.

• This can be useful for determining whether a set of linear equations has a unique solution.
• The determinant also affects eigenvalues, which are directly related to many phenomena in physics and engineering.

Understanding these properties helps in analyzing and solving computational problems efficiently.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that allows you to combine two matrices to produce a third matrix. This operation is not as straightforward as multiplying individual numbers, and there are specific rules involved.

• To multiply two matrices \( \mathbf{A} \) and \( \mathbf{B} \), the number of columns in \( \mathbf{A} \) must equal the number of rows in \( \mathbf{B} \).
• The resulting matrix's size is determined by the dimensions of \( \mathbf{A} \) and \( \mathbf{B} \): if \( \mathbf{A} \) is \( m \times n \) and \( \mathbf{B} \) is \( n \times p \), the result \( \mathbf{C} \) will be \( m \times p \).

Each element of the resulting matrix \( \mathbf{C} \) is computed as the dot product of the corresponding row in \( \mathbf{A} \) and column in \( \mathbf{B} \). This operation is crucial in various applications, such as computer graphics, physics simulations, and solving systems of equations.