Row and column transformations

[rsi]

I am stuck on this problem and keep going in a cycle coming back to the same state and would like to get hints on how to proceed. \( A \) is a \(R^{m*n} \) matrix and \( B \) is a \( R^{n*p} \) matrix. \( I_{n} \) is the \( n*n \) identity matrix.

Use elementary row and column operations to transform \[ \begin{bmatrix} I_{n} & 0 \\ 0 & AB \end{bmatrix} \] to \[ \begin{bmatrix} B & I_{n} \\ 0 & A \end{bmatrix} \].

 

[jvicens]

One approach you can take is to first focus on the bottom right quadrant of the matrix, which is the AB portion. Since you are trying to transform it into A, it would be helpful to try to get rid of the B portion. You can do this by using elementary row operations to swap the rows of the AB portion with the rows of the B portion. This will essentially move the B portion to the top right quadrant.

Once you have the B portion in the top right quadrant, you can then use elementary column operations to swap the columns of the AB portion with the columns of the A portion. This will move the A portion to the bottom right quadrant, completing the transformation.

Another approach is to use the fact that elementary row and column operations can be represented as matrix multiplication with elementary matrices. You can use this fact to create a sequence of elementary matrices that will transform the original matrix into the desired form. This approach may be more efficient and easier to visualize.

I hope these hints help you to find a solution to your problem. Good luck!