Linear Algebra

296 Chapter Four. Determinants(a)∣ 21 1 0−1 −1∣(b)3 0 2∣5 2 2∣2.9 For which values of k does this system have a unique solution?x + z − w = 2y − 2z = 3x + kz = 4z − w = 2̌ 2.10 Express each of these in terms of |H|.∣ h 3,1 h 3,2 h 3,3∣∣∣∣(a)h 2,1 h 2,2 h 2,3∣h 1,1 h 1,2 h 1,3∣ −h 1,1 −h 1,2 −h 1,3 ∣∣∣∣(b)−2h 2,1 −2h 2,2 −2h 2,3∣−3h 3,1 −3h 3,2 −3h 3,3∣ h 1,1 + h 3,1 h 1,2 + h 3,2 h 1,3 + h 3,3∣∣∣∣(c)h 2,1 h 2,2 h 2,3∣5h 3,1 5h 3,2 5h 3,3̌ 2.11 Find the determinant of a diagonal matrix.2.12 Describe the solution set of a homogeneous linear system if the determinantof the matrix of coefficients is nonzero.̌ 2.13 Show that this determinant is zero.y + z x + z x + yx y z∣1 1 1∣2.14 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by (−1) i+j .(b) Find the determinant of the square matrix with i, j entry (−1) i+j .2.15 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by i + j.(b) Find the determinant of the square matrix with i, j entry i + j.̌ 2.16 Show that determinant functions are not linear by giving a case where |A +B| ̸= |A| + |B|.2.17 The second condition in the definition, that row swaps change the sign of adeterminant, is somewhat annoying. It means we have to keep track of the numberof swaps, to compute how the sign alternates. Can we get rid of it? Can we replaceit with the condition that row swaps leave the determinant unchanged? (If so thenwe would need new 1 ×1, 2×2, and 3×3 formulas, but that would be a minormatter.)2.18 Prove that the determinant of any triangular matrix, upper or lower, is theproduct down its diagonal.2.19 Refer to the definition of elementary matrices in the Mechanics of MatrixMultiplication subsection.(a) What is the determinant of each kind of elementary matrix?(b) Prove that if E is any elementary matrix then |ES| = |E||S| for any appropriatelysized S.(c) (This question doesn’t involve determinants.) Prove that if T is singular thena product T S is also singular.(d) Show that |T S| = |T ||S|.(e) Show that if T is nonsingular then |T −1 | = |T | −1 .