Let be a natural number n≥2 and the n×n matrix whose entries at the i-th line and j-th column is min(i,j). Calculate:
a) its determinant
b) its inverse
Determinants are not altered by adding or subtracting one row from another.
The last row of the NxN matrix must have the ith column element equal to i since all of those elements will be <= N
The second to last row of the NxN matrix has all but the Nth column will have value i (the column number), and the Nth column itself will have a value of (N-1) since min(N-1, N) = N-1
Subtract row N-1 from row N. Then row N has zeros for the all columns except the Nth and 1 for the Nth column. Expanding the determinant on that row makes all terms vanish except the last, so
det NxN = det (N-1)*(N-1)
This recursive relation shows that all square matrices of this form have the same determinant. Since that determinant is 1 for the 2x2 case, it's 1 for all cases.
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Posted by Paul
on 2021-02-01 14:32:50 |
Just part (a)
