Let A=[O I B O] 2x2 matrix where all four submatrices are kxk. Determine A^2 and

Question

Let

A=[O I

B O] 2x2 matrix

where all four submatrices are kxk.

Determine A^2 and A^4.

At first, I thought the "O" was zero. It was pretty simple to find the answer but then I realize its the letter "O." Which got me confused. So, I mulitple A by A which I got the follow:

[O^2+BI 2IO

2BO BI+O^2]

I do not know where to go from here.

The answer is suppose to be [B O B O], 2x2 matrix, for A^2.

Could you please lead me in the right direction? Am I overthinking?

Thank you for your time and explanation.

Explanation / Answer

O is a null matrix, I is identity matrix Use the following identities, B*O = O*B = O (anything * null matrix is null matrix) B*I = I*B = B (any matrix * Identity matrix is that matrix) Therefore, A^2 = A* A = [O I * [O I = [ O*O+I*B O*I+I*O = [B O B O] B O] B*O+O*B B*I+O*O] O B] A^4 = A^2 * A^2 = [B O * [B O = [B^2 O O B] O B] O B^2]

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