Final answer:
Permutation matrices are used to rearrange a given matrix into upper or lower triangular forms. For the given matrix A, one permutation is needed to make it upper triangular by swapping first and third rows, while two permutations are required, one for rows and one for columns, to make it lower triangular.
Step-by-step explanation:
The question pertains to finding permutations that transform a given matrix A into upper triangular or lower triangular forms. An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. A lower triangular matrix is one where all the elements above the main diagonal are zero. To achieve this, one must use permutation matrices, which are special kinds of matrices that rearrange the rows or columns of another matrix when multiplied.
For the matrix A given as (0,0,6; 1,2,3; 0,4,5), a permutation matrix PA that makes it upper triangular would need to swap the first and third rows of A, because there is a non-zero element in the third row third column position, which is the appropriate location for the leading number in an upper triangular matrix.
For transforming matrix A into a lower triangular matrix P1AP2, two permutation matrices are required: P1 to rearrange the rows and P2 to rearrange the columns. The exact permutation matrices would depend on the initial arrangement of matrix A. However, a commonly used strategy would be to position elements on the diagonal in a descending order of magnitude, ensuring all elements above the diagonal are zero.