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Let A be an n×n matrix, I be n×n identity matrix. Define Lij =I+Mij, (1) i...

Let A be an n×n matrix, I be n×n identity matrix. Define
Lij =I+Mij, (1)
i > j, where the only non-zero element of Mij is mij on i-th row, j-th column.
1. Calculate LijA. What is the relationship between LijA and A?
2. Calculate L−1. ij
3. Suppose now we have a series of nonzero real numbers mi+1,i, mi+2,i, · · · , mn,i. Define Li+1,i , Li+2,i , · · · , Ln,i in the fashion of equation (1). Calculate the matrix product
Li+1,iLi+2,i · · · Ln,i.
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