108 CHAPTER 3. DETERMINANTS

andd (e1, · · · , en) = 1 (3.21)

where here ej is the vector in Fn which has a zero in every position except the jth

position in which it has a one.

3. Suppose f : Fn × · · · × Fn → F satisfies 3.20 and 3.21 and is linear in each variable.Show that f = d.

4. Show that if you replace a row (column) of an n × n matrix A with itself added tosome multiple of another row (column) then the new matrix has the same determinantas the original one.

5. Use the result of Problem 4 to evaluate by hand the determinant

det

1 2 3 2

−6 3 2 3

5 2 2 3

3 4 6 4

 .

6. Find the inverse if it exists of the matrix et cos t sin t

et − sin t cos t

et − cos t − sin t

 .

7. Let Ly = y(n) + an−1 (x) y(n−1) + · · · + a1 (x) y

′ + a0 (x) y where the ai are givencontinuous functions defined on an interval, (a, b) and y is some function which has nderivatives so it makes sense to write Ly. Suppose Lyk = 0 for k = 1, 2, · · · , n. TheWronskian of these functions, yi is defined as

W (y1, · · · , yn) (x) ≡ det

y1 (x) · · · yn (x)

y′1 (x) · · · y′n (x)...

...

y(n−1)1 (x) · · · y

(n−1)n (x)

Show that for W (x) =W (y1, · · · , yn) (x) to save space,

W ′ (x) = det

y1 (x) · · · yn (x)

... · · ·...

y(n−2)1 (x) y

(n−2)n (x)

y(n)1 (x) · · · y

(n)n (x)

 .

Now use the differential equation, Ly = 0 which is satisfied by each of these functions,yi and properties of determinants presented above to verify thatW ′+an−1 (x)W = 0.Give an explicit solution of this linear differential equation, Abel’s formula, and useyour answer to verify that the Wronskian of these solutions to the equation, Ly = 0either vanishes identically on (a, b) or never.

8. Two n × n matrices, A and B, are similar if B = S−1AS for some invertible n × nmatrix S. Show that if two matrices are similar, they have the same characteristicpolynomials. The characteristic polynomial of A is det (λI −A) .