2 . True of false , with reason if true and counterexaple if false: (15%) (a) if

Question

2 . True of false , with reason if true and counterexaple if false: (15%) (a) if A and B are identical except bi1-2ail, then detB-2detA (b) the determinant is the product of the pivots. (c) if A is invertible and B is singular, then A+B is invertible. (d) if A is invertible and B is singular, then AB is singular. (e) the determinant of AB-BA is zero. 3 . True of false, with reason if true and counterexapie it false: (15%) (a) for every matrix A, there is a solution to du/dt-Au starting from u(0) (b)every invertible matrix can be diagonalized (c)every diagonalizable matrix can be invertible. (d)exchanging the rows of 2by 2 matrix reverses the signs of its eigenvalues (e)if eigenvectors x and y corrspond to distinct eigenvalues, then x y-0.

Explanation / Answer

2. (a). False. If A =

1

2

4

2

3

5

3

2

7

then det(A) = -7 while det(B) = 4.

(b). False. If we scale any rows when getting the echelon form, we change the determinant.

(c ). True if A and B are of the same size/dimension. If A and B are both nxn matrices, A is invertible and B is singular, then A has linearly independent columns and B has linearly dependent columns so that A+B has linearly independent columns. Hence A+B is invertible.

(d). True if A and B are of the same size/dimension. If A and B are both nxn matrices, A is invertible and B is singular, then det(A) is non-zero, det(B) = 0. Also, det(AB) = det(A)det(B) = 0 so that AB is singular.

(e) False. Let A =

1

2

2

3

and B =

2

5

1

4

Then AB =

4

13

7

22

and BA =

12

19

9

14

so that AB-BA =

-8

-6

-2

8

Also, det(AB-BA) = -64-12= -76.

Please post the remaining question again.

1

2

4

2

3

5

3

2

7

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