Linear Algebra

330 Chapter Four. Determinants⎛ ⎞⎛ ⎞2 1 4 ( ) ( ) 1 4 3(a) ⎝−1 0 2⎠3 −1 1 1(b)(c)(d) ⎝−1 0 3⎠2 4 5 01 0 11 8 9̌ 1.17 Find the inverse of each matrix in the prior question with Theorem 1.9.1.18 Find the matrix adjoint to this one.⎛⎞2 1 0 0⎜1 2 1 0⎟⎝0 1 2 1⎠0 0 1 2̌ 1.19 Expand across the first row to derive the formula for the determinant of a 2×2matrix.̌ 1.20 Expand across the first row to derive the formula for the determinant of a 3×3matrix.̌ 1.21 (a) Give a formula for the adjoint of a 2×2 matrix.(b) Use it to derive the formula for the inverse.̌ 1.22 Can we compute a determinant by expanding down the diagonal?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 abovethe diagonal, that is, when i > j.(a) Must the adjoint of an upper triangular matrix be upper triangular? Lowertriangular?(b) Prove that the inverse of a upper triangular matrix is upper triangular, if aninverse 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 transposeusing Laplace’s expansion and induction on the size of the matrix.? 1.29 Show that1 −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]