Understanding the Matrix Problem: Finding A² + B²

We are given two matrices, A and B, with specific properties related to their multiplication:

• AB = B
• BA = A

Our goal is to find the value of the expression \(A^2 + B^2\).

To solve this, we first need to figure out what \(A^2\) and \(B^2\) are equal to, using the given conditions.

Calculating A² using Matrix Properties

The term \(A^2\) means A multiplied by itself, i.e., \(A \times A\). We can use the given conditions to simplify this expression.

\(A^2 = A \times A\)

We know that \(BA = A\). Let's substitute A on the right side of \(A \times A\) with \(BA\):

\(A^2 = A \times (BA)\)

Matrix multiplication is associative, meaning we can group the terms differently: \((A \times B) \times A\).

\(A^2 = (AB) \times A\)

We are given that \(AB = B\). Let's substitute \(AB\) with \(B\):

\(A^2 = (B) \times A\)

\(A^2 = BA\)

Finally, we know that \(BA = A\). So, we can substitute \(BA\) with \(A\):

\(A^2 = A\)

This shows that \(A^2\) is equal to A itself.

Calculating B² using Matrix Properties

Similarly, the term \(B^2\) means B multiplied by itself, i.e., \(B \times B\). We will use the given conditions AB = B and BA = A to simplify this expression.

\(B^2 = B \times B\)

We know that \(AB = B\). Let's substitute B on the right side of \(B \times B\) with \(AB\):

\(B^2 = B \times (AB)\)

Using the associative property of matrix multiplication: \((B \times A) \times B\).

\(B^2 = (BA) \times B\)

We are given that \(BA = A\). Let's substitute \(BA\) with \(A\):

\(B^2 = (A) \times B\)

\(B^2 = AB\)

Finally, we know that \(AB = B\). So, we can substitute \(AB\) with \(B\):

\(B^2 = B\)

This shows that \(B^2\) is equal to B itself.

Finding A² + B²

Now that we have found \(A^2\) and \(B^2\), we can calculate their sum:

\(A^2 + B^2\)

Substitute the values we found:

\(A^2 + B^2 = A + B\)

Therefore, \(A^2 + B^2\) is equal to \(A + B\).

Summary of Steps

Here's a quick look at the steps:

1. Start with the given conditions: \(AB = B\) and \(BA = A\).
2. Calculate \(A^2 = A \times A\). Use \(A = BA\) to write \(A^2 = A(BA) = (AB)A\). Use \(AB = B\) to get \((AB)A = BA\). Use \(BA = A\) to get \(BA = A\). Thus, \(A^2 = A\).
3. Calculate \(B^2 = B \times B\). Use \(B = AB\) to write \(B^2 = B(AB) = (BA)B\). Use \(BA = A\) to get \((BA)B = AB\). Use \(AB = B\) to get \(AB = B\). Thus, \(B^2 = B\).
4. Find \(A^2 + B^2 = A + B\).

The result \(A + B\) matches one of the given options.

Revision Table: Key Matrix Properties

Additional Information: Idempotent Matrices

The property we found, \(A^2 = A\) and \(B^2 = B\), is special. A matrix M is called an idempotent matrix if \(M^2 = M\).

In this problem, because \(A^2 = A\) and \(B^2 = B\), both matrices A and B are idempotent matrices under the given conditions \(AB=B\) and \(BA=A\).

Idempotent matrices are important in various areas of mathematics, including linear algebra, projection transformations, and statistics.

The conditions \(AB=B\) and \(BA=A\) imply that if A and B are square matrices, they must be idempotent.