Problem 37 - Advanced Engineering Mathematics

Question

Let \(\mathbf{A}=\left(\begin{array}{rr}3 & -4 \\ 1 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}7 & 4 \\ -1 & -5\end{array}\right)\). Verify that \(\operatorname{det}(\mathbf{A}+\mathbf{B}) \neq \operatorname{det} \mathbf{A}+\operatorname{det} \mathbf{B}\).

Step-by-Step Solution

Answer

\(\operatorname{det}(\mathbf{A} + \mathbf{B}) = -30\); \(\operatorname{det}(\mathbf{A}) + \operatorname{det}(\mathbf{B}) = -21\); they are not equal.

Step 1: Find Matrix A+B

First, we need to calculate the sum of matrices \(\mathbf{A}\) and \(\mathbf{B}\). Adding corresponding elements, we get: \[\mathbf{A} + \mathbf{B} = \begin{pmatrix} 3+7 & -4+4 \ 1+(-1) & 2+(-5) \end{pmatrix} = \begin{pmatrix} 10 & 0 \ 0 & -3 \end{pmatrix}.\]

Step 2: Calculate the Determinant of A+B

To find \(\operatorname{det}(\mathbf{A} + \mathbf{B})\), use the formula for the determinant of a 2x2 matrix: \[\operatorname{det}\begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc.\] Applying this to \(\mathbf{A} + \mathbf{B} = \begin{pmatrix} 10 & 0 \ 0 & -3 \end{pmatrix}\), we have: \[10 \times (-3) - 0 \times 0 = -30.\]

Step 3: Calculate the Determinant of A

For \(\mathbf{A} = \begin{pmatrix} 3 & -4 \ 1 & 2 \end{pmatrix}\), the determinant is \[3 \times 2 - (-4) \times 1 = 6 + 4 = 10.\]

Step 4: Calculate the Determinant of B

For \(\mathbf{B} = \begin{pmatrix} 7 & 4 \ -1 & -5 \end{pmatrix}\), the determinant is \[7 \times (-5) - 4 \times (-1) = -35 + 4 = -31.\]

Step 5: Verify the Determinant Relationship

We now compare the calculated determinants: \[\operatorname{det}(\mathbf{A} + \mathbf{B}) = -30.\] \[\operatorname{det}(\mathbf{A}) + \operatorname{det}(\mathbf{B}) = 10 - 31 = -21.\] Since \(-30 eq -21\), it is verified that \(\operatorname{det}(\mathbf{A} + \mathbf{B}) eq \operatorname{det}(\mathbf{A}) + \operatorname{det}(\mathbf{B})\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It is the process of adding two matrices by adding their corresponding elements. If you have two matrices \(\begin{pmatrix}a & b \c & d \end{pmatrix}\) and \(\begin{pmatrix}e & f \g & h \end{pmatrix}\), the sum is \(\begin{pmatrix}a+e & b+f \c+g & d+h \end{pmatrix}\).

• The matrices must have the same dimensions to be added.
• This operation is analogous to adding numbers, but done on an element-by-element basis.
• Matrix addition is both commutative and associative, meaning the order in which you add matrices doesn't affect their sum.

In our exercise, we explored the addition of matrices by combining elements of matrices \(\mathbf{A}\) and \(\mathbf{B}\), which both have dimensions 2x2, resulting in matrix \(\mathbf{A} + \mathbf{B} = \begin{pmatrix} 10 & 0 \ 0 & -3 \end{pmatrix}\).

2x2 Matrix Determinant

The determinant of a 2x2 matrix provides valuable insight into various matrix properties, including its invertibility. For a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant \( \operatorname{det} \), is calculated as: \[\operatorname{det} \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc. \]

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• A non-zero determinant implies the matrix is invertible.

In our calculation, the determinant of matrix \( \mathbf{A} + \mathbf{B} \) was found to be -30. Understanding the calculation process helps grasp why determinants are important in linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between them, including matrices and determinants. It provides the framework for solving systems of linear equations and transforming spaces.

• Key operations in linear algebra include matrix addition, multiplication, and finding determinants.
• It enables the study and manipulation of sets of equations or transformations that lend themselves to matrix formulation.
• Two major components in linear algebra are vectors and matrices, which can represent real-world systems in engineering, physics, and computer science.

Understanding linear algebra allows you to work seamlessly with multi-dimensional data and understand the nature of linear transformations.

Matrix Properties

Matrices hold a variety of properties that determine their behavior and possible operations on them. Here are a few fundamental matrix properties:

• Square Matrix: A matrix with the same number of rows and columns. Only square matrices possess determinants.
• Zero Matrix: A matrix with all elements being zero, often serving as the additive identity in matrix addition.
• Identity Matrix: A square matrix with ones on its diagonal and zeros elsewhere, acting as the multiplicative identity.
• Invertibility: A matrix is invertible if it has a non-zero determinant, allowing for the existence of its inverse, useful for solving equations.
• Symmetric Matrix: A matrix that is equal to its transpose, where \( A = A^T \).

Understanding these properties helps with solving equations, analyzing data, and performing computations in linear algebra.