If A and B are two matrices of order `3xx3` satisfying `AB=A and BA=B`, then `(A+B)^(5)`is equal to

Options

A. `5(A+B)`
B. `5I`
C. `16(A+B)`
D. `32I`

Answer

To solve the problem, we need to analyze the given conditions and derive the expression for \((A + B)^5\).

### Step 1: Understand the given conditions
We have two matrices \(A\) and \(B\) of order \(3 \times 3\) such that:
1. \(AB = A\)
2. \(BA = B\)

### Step 2: Analyze the implications of the conditions
From the equation \(AB = A\), we can rearrange it to \(AB - A = 0\) or \(A(B - I) = 0\). This suggests that \(A\) is a kind of projection matrix onto the column space of \(A\).

From the equation \(BA = B\), we can similarly rearrange it to \(BA - B = 0\) or \(B(A - I) = 0\). This suggests that \(B\) is a projection matrix onto the row space of \(B\).

### Step 3: Conclude about the matrices
Both conditions imply that \(A\) and \(B\) are idempotent matrices (i.e., \(A^2 = A\) and \(B^2 = B\)).

### Step 4: Find \(A + B\)
Now, we need to compute \((A + B)^5\). To do this, we first find \(A + B\):
- Since \(AB = A\) and \(BA = B\), we can deduce that:
- \(A + B\) behaves like a linear combination of identity matrices.

### Step 5: Calculate \((A + B)^5\)
Since both \(A\) and \(B\) are idempotent and satisfy the conditions provided, we can conclude that:
- \(A + B = I\) (the identity matrix) when \(A\) and \(B\) are projections onto complementary subspaces.

Thus, we can write:
\[ (A + B)^5 = (I)^5 = I \]

### Step 6: Final expression
Since \(I\) is the identity matrix, we can express the final result as:
\[ (A + B)^5 = 2^5 I = 32I \]

### Conclusion
The final answer is:
\[ (A + B)^5 = 32I \]