Problem 39 - Advanced Engineering Mathematics

Question

$$ (\mathbf{A}+\mathbf{B})^{2}=\mathbf{A}^{2}+2 \mathbf{A} \mathbf{B}+\mathbf{B}^{2} $$

Step-by-Step Solution

Answer

The expansion is correct: \((\mathbf{A} + \mathbf{B})^2 = \mathbf{A}^2 + 2\mathbf{A}\mathbf{B} + \mathbf{B}^2\).

Step 1: Identify the Problem

The exercise asks us to confirm the expansion of \((\mathbf{A}+\mathbf{B})^2\), comparing it with the expression provided: \(\mathbf{A}^2 + 2\mathbf{A}\mathbf{B} + \mathbf{B}^2\).

Step 2: Apply Binomial Theorem

According to the binomial theorem, the expansion of \((\mathbf{A} + \mathbf{B})^2\) is \((\mathbf{A} + \mathbf{B}) \times (\mathbf{A} + \mathbf{B})\), which leads to the calculation as \(\mathbf{A}\mathbf{A} + \mathbf{A}\mathbf{B} + \mathbf{B}\mathbf{A} + \mathbf{B}\mathbf{B}\).

Step 3: Combine Like Terms

In the expansion, the term \(\mathbf{A}\mathbf{A}\) becomes \(\mathbf{A}^2\), \(\mathbf{B}\mathbf{B}\) becomes \(\mathbf{B}^2\), and the terms \(\mathbf{A}\mathbf{B} + \mathbf{B}\mathbf{A}\) combine to form \(2\mathbf{A}\mathbf{B}\).

Step 4: Validate by Comparison

Compare the expanded form \(\mathbf{A}^2 + 2\mathbf{A}\mathbf{B} + \mathbf{B}^2\) with the right side of the equation. They are identical, verifying the provided statement.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool that expands expressions raised to a power. When working with matrices, the theorem helps to expand expressions like \( ( extbf{A}+ extbf{B})^{2} \), where \( extbf{A} \) and \( extbf{B} \) are matrices. In basic algebra, the binomial theorem for two numbers \( a \) and \( b \) can be expressed as:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

Similarly, the theorem can be applied to matrices, allowing us to expand:

• \( ( extbf{A} + extbf{B})^2 = extbf{A}^2 + extbf{A} extbf{B} + extbf{B} extbf{A} + extbf{B}^2 \)

Note that matrix multiplication is not commutative, meaning \( extbf{A} extbf{B} \) is not necessarily equal to \( extbf{B} extbf{A} \). Therefore, if \( extbf{A} \) and \( extbf{B} \) do not commute, the expression \( extbf{A} extbf{B} + extbf{B} extbf{A} = 2 extbf{A} extbf{B} \) is not simplifiable unless specific conditions are met, such as \( extbf{A} \) and \( extbf{B} \) being matrices of specific forms like identities or similar matrices.

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. It combines two matrices to produce a third one, following specific rules. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second. The resulting matrix dimensions are determined by the rows of the first matrix and the columns of the second matrix.Key properties include:

• Non-commutative: \( extbf{A} extbf{B} eq extbf{B} extbf{A} \) in general.
• Associative: \( ( extbf{A} extbf{B}) extbf{C} = extbf{A}( extbf{B} extbf{C}) \)
• Distributive: \( extbf{A}( extbf{B} + extbf{C}) = extbf{A} extbf{B} + extbf{A} extbf{C} \)

During the process of expanding \( ( extbf{A} + extbf{B})^2 \), each combination of matrices \( extbf{A} \) and \( extbf{B} \) was multiplied by following these properties. Each term was carefully calculated to eventually simplify into the recognizable form of the binomial expansion, demonstrating the blend between algebraic operations and matrix characteristics.

Matrix Expansion

The concept of matrix expansion involves breaking down a complex matrix expression into simpler components that are easier to understand and calculate. In the case of \( ( extbf{A}+ extbf{B})^{2} \), expansion helps to reveal all possible products of matrices involved.To expand a matrix expression like \( ( extbf{A} + extbf{B}) imes ( extbf{A} + extbf{B}) \), you follow these steps:

• Multiply each term in the first matrix expression by each term in the second.
• Combine terms that can be simplified, taking care of matrix operation rules.

The expansion of \( ( extbf{A}+ extbf{B})^{2} \) thus involves evaluating \( extbf{A} extbf{A}, extbf{A} extbf{B}, extbf{B} extbf{A}, extbf{B} extbf{B} \), combining like terms, and respecting the unique properties of matrix operations.This technique is pivotal in translating complex expressions into simpler sums of their matrix products, simplifying analysis and further applications in more advanced mathematics.