Simplify Matrix Expression with \(A^2 = A\) Property

The problem asks us to simplify the matrix expression \(A(I - 2A)^3 + 2A^3\) given that A is a square matrix, I is the identity matrix, and \(A^2 = A\).

Understanding the Given Matrix Property: \(A^2 = A\)

The condition \(A^2 = A\) means that the matrix A is idempotent. This property is very useful in simplifying expressions involving powers of A.

If \(A^2 = A\), we can find higher powers of A:

• \(A^3 = A \cdot A^2 = A \cdot A = A^2 = A\)
• \(A^4 = A \cdot A^3 = A \cdot A = A^2 = A\)

In general, for an idempotent matrix A, \(A^n = A\) for any positive integer \(n \ge 1\).

Expanding the Term \((I - 2A)^3\)

We need to expand \((I - 2A)^3\). We can do this step by step using matrix multiplication or by recalling the binomial expansion formula \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) and applying matrix properties (\(I\) acts like 1 in multiplication, \(IA = A\), \(AI = A\), \(I^n = I\)).

Let's expand \((I - 2A)^2\) first:

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

\(= I \cdot I - I \cdot (2A) - (2A) \cdot I + (2A) \cdot (2A)\)

\(= I^2 - 2IA - 2AI + 4A^2\)

Using \(I^2 = I\), \(IA = A\), \(AI = A\):

\(= I - 2A - 2A + 4A^2\)

\(= I - 4A + 4A^2\)

Now, use the given property \(A^2 = A\):

\(= I - 4A + 4A\)

\(= I\)

So, \((I - 2A)^2 = I\). This is a significant simplification!

Now, let's find \((I - 2A)^3\):

\((I - 2A)^3 = (I - 2A)^2 (I - 2A)\)

Since \((I - 2A)^2 = I\), we have:

\(= I(I - 2A)\)

\(= I \cdot I - I \cdot (2A)\)

\(= I^2 - 2IA\)

Using \(I^2 = I\) and \(IA = A\):

\(= I - 2A\)

Thus, \((I - 2A)^3 = I - 2A\).

Simplifying the Main Expression \(A(I - 2A)^3 + 2A^3\)

Now we substitute the simplified \((I - 2A)^3\) and \(A^3\) (which is A) back into the original expression:

Original expression: \(A(I - 2A)^3 + 2A^3\)

Substitute \((I - 2A)^3 = I - 2A\): \(A(I - 2A) + 2A^3\)

Substitute \(A^3 = A\): \(A(I - 2A) + 2A\)

Now, expand \(A(I - 2A)\):

\(A(I - 2A) = A \cdot I - A \cdot (2A)\)

\(= AI - 2A^2\)

Using \(AI = A\):

\(= A - 2A^2\)

Now, use the given property \(A^2 = A\):

\(= A - 2A\)

\(= -A\)

So, \(A(I - 2A)^3\) simplifies to \(-A\).

Substitute this back into the expression \(A(I - 2A)^3 + 2A\):

\(= -A + 2A\)

\(= A\)

The simplified expression is A.

Conclusion

Given that A is a square matrix and I is an identity matrix such that \(A^2 = A\), the expression \(A(I - 2A)^3 + 2A^3\) simplifies to A.

Revision Table: Key Matrix Properties Used

Additional Information on Idempotent Matrices

An idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, \(A^2 = A\). These matrices appear in various areas of linear algebra and statistics, particularly in regression analysis and projection operators.

Some properties of idempotent matrices A (where A is an n x n matrix):

• The eigenvalues of an idempotent matrix are either 0 or 1.
• The determinant of an idempotent matrix is either 0 or 1.
• If A is idempotent, then \((I - A)\) is also idempotent. Let's check: \((I-A)^2 = (I-A)(I-A) = I^2 - IA - AI + A^2 = I - A - A + A = I - A\). This confirms \((I-A)\) is idempotent.
• If A is idempotent, then \(A^T\) (transpose) is also idempotent.

Understanding these basic matrix properties and definitions is crucial for solving matrix algebra problems like this one.