10 True or False a T F The value of a cofactor expansion of

10. True or False. (a) T F: The value of a cofactor expansion of a matrix A is independent of the row or column chosen for the expansion. (b) T F: For any square matrices A and B, det(A + B)= det(A) + det(B). (c) T F: The minor Mij of a square matrix is equal to its cofactor Cij if and only if i + j is even. (d) T F: If A is an n x n matrix with n > = 2 and B is the matrix obtained from A by inter-changing the first two rows and then interchanging the last two rows, then det(B) = det(A). (e) T F: If A is a square matrix and B is the matrix obtained from A by multiplying one of the rows by 3, then det(B) = 3det(A). (f) T F: If A is an n x n matrix with n > = 3 and B is the matrix obtained from A by adding 7 times the first row to the second row and then adding 5 times the first row to the last, then det(B) = 35 det(A). (g) T F: If a square matrix A has two identical rows, then det(A) = 0. (h) T F: If det(A) = 0 for a square matrix, then A should have two identical rows. (i) T F: If a square matrix A has two identical columns, then det(A) = 0. (j) T F: If one row of a square matrix A is a constant multiple of another, then det(A) = 0.

Solution

a) T

b) F

c) T

d) T

e) T

f) F

g) T

h) F

i) T

j) T