Simplifying Matrix Expressions with Involuntary Matrices

This solution explains how to simplify the matrix expression $(I + A)^2 - (I - A)^2$. We are given that $A$ is an involuntary matrix and $I$ is the unit matrix (also known as the identity matrix) of the same order. Understanding the properties of these specific matrices is key to solving this problem.

Understanding Involuntary and Unit Matrices

Before diving into the calculation, let's clarify the definitions and properties of the matrices involved:

• Unit Matrix (I): The unit matrix, typically denoted by $I$, is the identity matrix. Its main diagonal consists of ones and all other entries are zero. Key properties include:
  • $I^2 = I$ (Multiplying the identity matrix by itself yields the identity matrix).
  • $AI = IA = A$ (Multiplying any matrix $A$ by the identity matrix $I$ results in the matrix $A$ itself).
• Involuntary Matrix (A): A square matrix $A$ is called an involuntary matrix if its square is equal to the unit matrix $I$. The defining property is:
  • $A^2 = I$

Step-by-Step Calculation of the Expression $(I + A)^2 - (I - A)^2$

The goal is to simplify the expression $(I + A)^2 - (I - A)^2$. We can achieve this by expanding each squared term individually using the rules of matrix multiplication and the properties defined above.

Expanding the First Term: $(I + A)^2$

We expand $(I + A)^2$ using the distributive property, similar to algebraic expansion:

$(I + A)^2 = (I + A)(I + A)$

Applying the distributive law:

= $I(I + A) + A(I + A)$

= $I \cdot I + I \cdot A + A \cdot I + A \cdot A$

Now, we use the properties of the unit matrix ($I^2 = I$, $IA = A$, $AI = A$) and the involuntary matrix ($A^2 = I$):

= $I + A + A + I$

Combining like terms:

= $2I + 2A$

Expanding the Second Term: $(I - A)^2$

Similarly, we expand the second term $(I - A)^2$:

$(I - A)^2 = (I - A)(I - A)$

Applying the distributive law:

= $I(I - A) - A(I - A)$

= $I \cdot I - I \cdot A - A \cdot I + A \cdot A$

Using the same properties ($I^2 = I$, $IA = A$, $AI = A$, $A^2 = I$):

= $I - A - A + I$

Combining like terms:

= $2I - 2A$

Subtracting the Expanded Terms

Now, we subtract the expanded form of the second term from the expanded form of the first term:

$(I + A)^2 - (I - A)^2 = (2I + 2A) - (2I - 2A)$

Distribute the negative sign to the terms in the second parenthesis:

= $2I + 2A - 2I + 2A$

Rearrange and group the terms involving $I$ and $A$:

= $(2I - 2I) + (2A + 2A)$

Perform the addition and subtraction:

= $0 + 4A$

= $4A$

Conclusion: The Simplified Expression $4A$

Through careful expansion and application of the properties of involuntary ($A^2 = I$) and unit ($I^2 = I$, $AI = IA = A$) matrices, we have shown that the expression $(I + A)^2 - (I - A)^2$ simplifies precisely to $4A$.