- This exercise is recommended for all readers.
- This exercise is recommended for all readers.
- Problem 3
Find the adjoint of the matrix in Example 1.6.
- Answer
- This exercise is recommended for all readers.
- Problem 4
Find the matrix adjoint to each.
-
-
-
-
- Answer
-
- The minors are
:
.
-
-
- This exercise is recommended for all readers.
- Problem 6
Find the matrix adjoint to this one.

- Answer
- This exercise is recommended for all readers.
- Problem 7
Expand across the first row to derive the formula for the determinant
of a
matrix.
- Answer
The determinant

expanded on the first row gives
(note the two
minors).
- This exercise is recommended for all readers.
- Problem 8
Expand across the first row to derive the formula for the determinant
of a
matrix.
- Answer
The determinant of

is this.

- This exercise is recommended for all readers.
- Problem 9
- Give a formula for the adjoint of a
matrix.
- Use it to derive the formula for the inverse.
- Answer
-
-
- This exercise is recommended for all readers.
- Problem 10
Can we compute a determinant by expanding down the diagonal?
- Answer
No.
Here is a determinant whose value

doesn't equal the result of
expanding down the diagonal.

- Problem 11
Give a formula for the adjoint of a diagonal matrix.
- Answer
Consider this diagonal matrix.

If
then the
minor is an
matrix
with only
nonzero entries, because both
and
are
deleted.
Thus, at least one row or column of the minor is all zeroes, and
so the cofactor
is zero.
If
then the minor is the diagonal matrix with entries
, ...,
,
, ...,
.
Its determinant is obviously
times the product of those.

By the way, Theorem 1.9 provides a slicker way to derive this conclusion.
- This exercise is recommended for all readers.
- Problem 13
Prove or disprove:
.
- Answer
It is false; here is an example.

- Problem 14
A square matrix is upper triangular if
each
entry is zero in the part above the diagonal,
that is, when
.
-
Must the adjoint of an upper triangular matrix be upper triangular?
Lower triangular?
- Prove that the inverse of a upper triangular matrix
is upper triangular, if an inverse exists.
- Answer
- An example

suggests the right answer.

The result is indeed upper triangular.
A check of this is detailed but not hard.
The entries in the upper triangle of the adjoint are
where
.
We need to verify that the cofactor
is zero if
.
With
, row
and column
of
,

when deleted, leave an upper triangular minor,
because entry
of the minor is either entry
of
(this happens if
and
;
in this case
implies that the entry is zero)
or it is entry
of
(this happens if
and
; in this case,
implies that
, which implies
that the entry is zero), or it is entry
of
(this last case happens when
and
; obviously here
implies that
and so the entry is zero).
Thus the determinant of the minor is the product down the
diagonal.
Observe that the
entry of
is the
entry of the minor (it doesn't get
deleted because the relation
is strict).
But this entry is zero because
is upper triangular and
.
Therefore the cofactor is zero, and the adjoint is upper triangular.
(The lower triangular case is similar.)
- This is immediate from the prior part, by
Corollary 1.11.
- Problem 15
This question requires material from the optional Determinants Exist subsection.
Prove Theorem 1.5
by using the permutation expansion.
- Answer
We will show that each determinant can be expanded along
row
.
The argument for column
is similar.
Each term in the permutation expansion contains one and
only one entry from each row.
As in Example 1.1,
factor out each row
entry to get
,
where each
is a sum of terms not containing any
elements of row
.
We will show that
is the
cofactor.
Consider the
case first:

where the sum is over all
-permutations
such that
.
To show that
is the minor
, we need only show
that if
is an
-permutation such that
and
is an
-permutation with
, ...,
then
.
But that's true because
and
have the same number of inversions.
Back to the general
case. Swap adjacent rows until the
-th is last and swap adjacent columns until the
-th is last. Observe that the determinant of the
-th minor is not affected by these adjacent swaps because inversions are preserved (since the minor has the
-th row and
-th column omitted). On the other hand, the sign of
and
is changed
plus
times. Thus
.
- Problem 16
Prove that the determinant of a matrix equals the determinant of its
transpose using Laplace's expansion and induction on the size
of the matrix.
- Answer
This is obvious for the
base case.
For the inductive case, assume that the determinant of a matrix equals the determinant of its transpose for all
, ...,
matrices. Expanding on row
gives
and expanding on column
gives
Since
the signs are the same in the two summations. Since the
minor of
is the transpose of the
minor of
, the inductive hypothesis gives
.
- ? Problem 17
Show that

where
is the
-th term of
, the Fibonacci sequence,
and the determinant is of order
.
(Walter & Tytun 1949)
- Answer
This is how the answer was given in the cited source.
Denoting the above determinant by
, it is seen that
,
.
It remains to show that
.
In
subtract the
-th column from the
-th,
the
-th from the
-th, ..., the first from
the third, obtaining

By expanding this determinant with reference to the first row, there results the desired relation.
- Walter, Dan (proposer); Tytun, Alex (solver) (1949), "Elementary problem 834", American Mathematical Monthly, American Mathematical Society, 56 (6): 409.