Determinants
Part 3: The Adjoint
If A is an n x n matrix and i and j are indices
between 1 and n, we denote by Ai,j the
n - 1 by n - 1 matrix obtained from A by deleting
the ith row and the jth column of A.
The (i,j)-cofactor of A is the number
(-1)i+jdet(Ai,j). The
cofactor matrix of A is the n x
n matrix whose (i,j)th entry is the (i,j)-cofactor
of A. The classical adjoint of A,
denoted adj(A), is the transpose of the cofactor matrix of A.
- Before you ask your
computer algebra system to compute any adjoints,
carry out the following experiment to be sure you understand the definition.
Compute the (3,2) entry of the adjoint of the matrix A (already defined
in the worksheet) by setting up an appropriate expression and calculating its value. Check your work by entering adj(A).
- Compute A*adj(A), B*adj(B),
and T*adj(T). What
do you observe?
- Suppose M is an invertible matrix. Use your
result in the preceding step to find a formula for the inverse of M that
uses adj(M). Use this formula to define a matrix V that should be
the inverse of A. Test your formula by computing VA and AV (or by computing
one of these and explaining why you don't need both).
- The result of Step 3 can be used to construct
a non-trivial matrix that has an inverse with integer entries. (There
are much more important things we can do with this formula, but this exercise
will reveal the source of all those artificial-looking textbook inversion
problems.) Start by entering a 5 x 5 upper triangular matrix N whose entries
are all integers and whose determinant is 1 or -1. (If necessary, refer
to Step 5 in Part 2.) Now apply row replacement operations to change N until
it has mostly non-zero entries. You can also apply column replacement operations in the same way. Continue until there is no discernible pattern in your matrix. Explain why
the inverse of the matrix you created must have only integer entries. Verify
this by computing the inverse.
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