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MATH-4100
Linear Algebra: Elementary matrices and
exact inversion of integer matrices.
Created by Professor M. Zuker
 
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Background

For a positive integer, n, we will define three kinds of elementary n × n matrices. In the following, c is a scalar and j, k are integers between 1 and n, inclusive.
  1. If j ≠k let En(j,k,c) be the n × n identity matrix modified by placing c in row j and column k.
  2. For any non-zero scalar, c, let En(j,j,c) be the n × n diagonal matrix containing c in the jth diagonal position and 1 elsewhere.
  3. If j ≠k let Πn(j,k) be the n × n identity matrix where rows j and k have been interchanged. (Interchanging colums j and k on the identity matrix produces the same result.)
Note that the inverse of En(j,k,c) is En(j,k,-c) if j ≠k, the inverse of En(j,j,c) is En(j,j,1/c) and that Πn(j,k) is its own inverse. If A is an m × n matrix (m rows and n columns), then
  1. For any scalar, c, adding c × row k to row j of A is equivalent to multiplying A on the left by Em(j,k,c), j ≠k.
  2. For any non-zero scalar, c, multiplying row j of A by c is equivalent to multiplying A on the left by Em(j,j,c).
  3. If j ≠k, interchanging (swapping) rows j and k is equivalent to multiplying A on the left by Πm(j,k).
Also,
  1. For any scalar, c, adding c × column k to column j of A is equivalent to multiplying A on the right by En(k,j,c), j ≠k. Note that k and j have swapped positions from the above.
  2. For any non-zero scalar, c, multiplying column j of A by c is equivalent to multiplying A on the right by En(j,j,c).
  3. If j ≠k, interchanging (swapping) columns j and k is equivalent to multiplying A on the right by Πn(k,j). (Πn(k,j) = Πn(j,k). )

Enter a square matrix in the box below. Use integers only. The number of entries in each row must equal the total number of rows.   Example: Inversion of an 8×8 matrix containing the first 64 prime numbers.

   

Michael Zuker
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
2008-09-24