Linear Algebra, 2020a

368 Chapter Four. Determinants
1.23 Give a formula for the adjoint of a diagonal matrix.
̌ 1.24 Prove that the transpose of the adjoint is the adjoint of the transpose.
1.25 Prove or disprove: adj(adj(T)) = T.
1.26 A square matrix is upper triangular if each i, j entry is zero in the part above
the diagonal, that is, when i>j.
(a) Must the adjoint of an upper triangular matrix be upper triangular? Lower
triangular?
(b) Prove that the inverse of a upper triangular matrix is upper triangular, if an
inverse exists.
1.27 This question requires material from the optional Determinants Exist subsection.
Prove Theorem 1.5 by using the permutation expansion.
1.28 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.
? 1.29 Show that
1 −1 1 −1 1 −1 ...
1 1 0 1 0 1 ...
F n =
0 1 1 0 1 0 ...
0 0 1 1 0 1 ...
∣. . . . . . ... ∣
where F n is the n-th term of 1,1,2,3,5,...,x,y,x+ y,..., the Fibonacci sequence,
and the determinant is of order n − 1. [Am. Math. Mon., Jun. 1949]