Problems for Section 9.3

In Problems 1-7, below, D represents a determinant of order n. Prove each statement either from the definition of an n by n determinant, by using the Transpose Theorem, or by using previous results.

R 1. If all the entries on a given row (or column) of D are multiplied by a fixed number k, the value of D is multiplied by k.

R 2. If each entry in a given row (or column) of D is zero, then D = 0.

R 3. (a) If any two columns of a determinant D are interchanged, the resulting determinant D₁ equals -D. (See Problem 21, Chapter 7.)

(b) If any two rows of a determinant D are interchanged, the resulting determinant D₂ equals -D.

R 4. If the entries of a row (or column) of D are a constant k times the corresponding entries of another row (or column), then D = 0.

R 5. If the entries of a given row (or column) of D are f₁ + g₁, f₂ + g₂, ..., f_n + g_n, then
D = D₁ + D₂, where D₁ results from D by replacing the given row (or column) by f₁, f₂, ..., f_n and D₂ by replacing the given row (or column) by g₁, g₂, ..., g_n.

R 6. Let u₁, u₂, ..., u_n and v₁, v₂, ..., v_n be the entries of two rows (or columns) of D, and let D^* result from replacing v₁, v₂, ..., v_n in D by v₁ + ku₁, v₂ + ku₂, ..., v_n + ku_n, respectively.
Then D^* = D.

R 7. Let a_ij be the entry in the ith row and jth column of D.

(a) If S is the sum of all the terms of the expansion of D that involve a_nn, then S = a_nnM_nn = a_nnC_nn, where M_nn and C_nn are the minor and cofactor of a_nn. (See Problem 22, Chapter 7.)

(b) Let T be the sum of all the terms of the expansion of D that involve a fixed entry a_hk. Then
T = (-1)^h+ka_hkM_hk = a_hkC_hk. (Use Problem 3, above, and Part (a) of this problem.)

(c) If h is one of the numbers 1, 2, ...., n, then each term of the expansion of D has one and only one of the entries a_h1, a_h2, ..., a_hn as a factor.

(d) 

(e)

(f) If k is any one of the numbers 1, 2, ..., n then 
 

Tuesday, June 23, 1998