Key Concepts and Formulas

This problem primarily tests your understanding of matrix algebra, particularly properties of orthogonal matrices and expansion of matrix expressions.

1. Orthogonal Matrix Definition: A square matrix $A$ is called an orthogonal matrix if $A A^T = I$, where $A^T$ is the transpose of $A$ and $I$ is the identity matrix. A key property of orthogonal matrices is that $A A^T = I$ implies $A^T A = I$. This means an orthogonal matrix is always invertible, and its inverse is its transpose ($A^{-1} = A^T$).
2. Matrix Multiplication Commutativity: In general, for two matrices $X$ and $Y$, $XY \neq YX$. However, if $XY=YX$, we say $X$ and $Y$ commute. For an orthogonal matrix $A$, $A$ and $A^T$ commute because $A A^T = A^T A = I$. This is crucial for expanding terms like $(A+A^T)^2$.
3. Algebraic Identities for Commuting Matrices: If $X$ and $Y$ are matrices that commute ($XY=YX$), then the standard algebraic identities apply:
   • $(X+Y)^2 = X^2 + XY + YX + Y^2 = X^2 + 2XY + Y^2$
   • $(X-Y)^2 = X^2 - XY - YX + Y^2 = X^2 - 2XY + Y^2$

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Step-by-Step Derivation

Step 1: Understand the given condition and its implications. We are given that $A$ is a square matrix such that $A A^T = I$.

• Explanation: This condition immediately tells us that $A$ is an orthogonal matrix. A fundamental property of orthogonal matrices is that if $A A^T = I$, then it must also be true that $A^T A = I$.
• Implication: Since $A A^T = I$ and $A^T A = I$, it means that $A$ and $A^T$ commute with each other. This is a very important detail for the subsequent algebraic expansions.

Step 2: Simplify the expression inside the brackets. Let's first focus on simplifying the expression $\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$.

• Explanation: We will use the algebraic identities for commuting matrices derived earlier. Since $A$ and $A^T$ commute, we can apply the standard $(X+Y)^2$ and $(X-Y)^2$ formulas.
• Expansion: $(A+A^T)^2 = A^2 + A A^T + A^T A + (A^T)^2$ $(A-A^T)^2 = A^2 - A A^T - A^T A + (A^T)^2$ Now, substitute $A A^T = I$ and $A^T A = I$ into these expansions: $(A+A^T)^2 = A^2 + I + I + (A^T)^2 = A^2 + 2I + (A^T)^2$ $(A-A^T)^2 = A^2 - I - I + (A^T)^2 = A^2 - 2I + (A^T)^2$
• Combining the terms: Now, add these two expanded expressions: $\left(A+A^T\right)^2+\left(A-A^T\right)^2 = \left(A^2 + 2I + (A^T)^2\right) + \left(A^2 - 2I + (A^T)^2\right)$ $= A^2 + 2I + (A^T)^2 + A^2 - 2I + (A^T)^2$
• Simplification: Observe that the terms $+2I$ and $-2I$ cancel each other out. $= 2A^2 + 2(A^T)^2$

Step 3: Substitute the simplified expression back into the original problem. Now, substitute $2A^2 + 2(A^T)^2$ back into the original expression: $\frac{1}{2} A\left[2A^2 + 2(A^T)^2\right]$

• Explanation: We can factor out the scalar $2$ from the bracketed term.
• Simplification: $= \frac{1}{2} A \cdot 2 \left[A^2 + (A^T)^2\right]$ $= A \left[A^2 + (A^T)^2\right]$

Step 4: Distribute $A$ and perform final simplification. Distribute the matrix $A$ into the bracketed term: $A \cdot A^2 + A \cdot (A^T)^2$ $= A^3 + A(A^T A^T)$

• Explanation: We need to simplify the term $A(A^T A^T)$. Matrix multiplication is associative, meaning $(XY)Z = X(YZ)$.
• Simplification: $A(A^T A^T) = (A A^T) A^T$ Now, use the given condition $A A^T = I$: $(A A^T) A^T = I A^T$ Since $I$ is the identity matrix, multiplying any matrix by $I$ results in the same matrix: $I A^T = A^T$
• Final Result: Substituting this back into our expression: $A^3 + A^T$

Thus, the given expression simplifies to $A^3 + A^T$.

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Tips for Success and Common Mistakes to Avoid

• Recognize Orthogonal Matrices: Always identify if a matrix is orthogonal ($AA^T = I$). This immediately provides the crucial property $A^T A = I$, which simplifies many calculations.
• Matrix Commutativity is Key: When expanding expressions like $(X+Y)^2$ for matrices, remember that $XY$ is generally not equal to $YX$. You must write it as $X^2 + XY + YX + Y^2$. Only if you know $X$ and $Y$ commute (as $A$ and $A^T$ do here), can you simplify to $X^2 + 2XY + Y^2$. Failing to consider commutativity is a very common mistake.
• Associativity of Matrix Multiplication: Remember that $(AB)C = A(BC)$. This is vital when simplifying terms like $A(A^T)^2$. Don't just cancel terms without careful application of properties. For example, $A(A^T)^2$ is not $A^2 A^T$ unless $A$ and $A^T$ commute.
• Identity Matrix Properties: $IX = X$ and $XI = X$ for any matrix $X$ (of compatible dimensions). This is used in the final simplification.

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Summary and Key Takeaway

This problem beautifully demonstrates the importance of understanding the fundamental properties of matrices, especially orthogonal matrices. By recognizing $A A^T = I$ implies $A^T A = I$, we establish that $A$ and $A^T$ commute, allowing for standard algebraic expansions. The subsequent simplification relies on careful application of matrix associativity and the identity matrix property. Mastering these concepts is crucial for solving complex problems in matrix algebra.

The final answer is $\boxed{\mathrm{A}^3+\mathrm{A}^{\mathrm{T}}}$.