Are there two 3 by 3 matrices A and B, both of rank 2,such that AB is the zero matrix? Explain your answer

Yes, there are two 3 by 3 matrices A and B, both of rank 2, such that AB is the zero matrix. Here is the explanation:

Step 1: Understand the Rank of a Matrix The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. A 3 by 3 matrix of rank 2 means that there are only 2 linearly independent rows or columns, and the third one is a linear combination of the other two.

Step 2: Construct Matrices A and B Let's construct two 3 by 3 matrices A and B, both of rank 2. For simplicity, let's make the third row of both matrices a linear combination of the first two rows.

Matrix A: 1 0 0 0 1 0 1 1 0

Matrix B: 1 0 0 0 1 0 1 1 0

Step 3: Multiply Matrices A and B When we multiply matrices A and B, we get the zero matrix. This is because the third row in both matrices is a linear combination of the first two rows, and when these rows are multiplied, they result in zero.

So, yes, it is possible to have two 3 by 3 matrices A and B, both of rank 2, such that AB is the zero matrix.