Problems on Chapter 6: Determinants Problems on Chapter 6

1. Show that det 2AAA  det   det  .

2. By using determinant, find the values of k so that the following vectors do not form a basis for 3 : k  1   1        1  , k  ,  1        1   1  k 

3. Let A and B be 6 by 6 matrices and detA   2 , detB  3 . Find

5 (a) det A3  (b) det 5B (c) det AB  (d) det AB1  1 

1001 2001 3001    4. Find det A where A  1002 2002 3002 .   1003 2003 3003 

5. Find the determinant of the following matrix without using a calculator: 1 2 0 0    3 4 0 0 A    6 7 1 8    2 1 3 4 

n 6. Let e1,,, e 2  en be our standard basis for  . Show that

det  x1e 1 x 2 e 2 xn e n  x 1 x 2  x n

7. Show that the transformation TM: nn   given by T AA  det   where A is a n by n matrix is not linear.

8. Prove the following result: n detc1 c 2 cn  0   c 1 , c 2 ,  , c n form a basis for 

9. Determine the following shaded area: 1 Page Problems on Chapter 6

Figure 1

10. The equation of a line through two points P  x1, y 1  and Q  x2, y 2  is given by x y 1     detx1 y 1 1  0   x2 y 2 1  Find the equation of the line through the points 5, 6 and  6, 5 and sketch these points and the line on a graph.

0 1 0 0    0 0 1 0 11. Find det A where A    . 0 0 0 1    1 0 0 0 

a b 0 0 0    0a b 0 0  12. Find det A where A  0 0a b 0  .   0 0 0 a b    b0 0 0 a 

13. Let An be the n by n matrix with 1’s along the secondary diagonal and zeros 0 0 1     everywhere else in the matrix. For example A3 0 1 0  .   1 0 0 

(i) Find the determinants of AAAAA2,,,, 3 4 5 6 . 2 (ii) Predict a formula for det An  . Page Problems on Chapter 6 13 4  14. Let A    . Determine the matrix X given by 1 1  1 XAAI det        traceAA  2 det   Where the trace of the matrix is sum of all the leading terms and I is the identity matrix. Also find X2 . What do you notice about your result?

1 1 1 1 1  1 0 0 0 0      0 1 2 3 4  1 1 0 0 0  15. Let U  0 0 1 3 6  and L  1 2 1 0 0  . The Pascal matrix P is a     5 0 0 0 1 4  1 3 3 1 0      0 0 0 0 1  1 4 6 4 1  matrix whose entries are the binomial coefficients of an expansion to the index 5.

1 1 1 1 1    1 2 3 4 5  (i) Show that P LU  1 3 6 10 15  . 5   1 4 10 20 35    1 5 15 35 70 

(ii) Find the determinants of LUP, and 5 .

16. Find a non-zero 3 by 3 matrix whose determinant is zero.

17. Either proof or provide a counter example of the following: If A is a non-zero triangular matrix then A is invertible.

18. Show that for same size square matrices A and B: (i) detAB BA  0 (ii) detAB  det BA  0

19. Is any set of square matrices whose determinant is zero a vector space with respect to the usual matrix addition and scalar multiplication?

20. Prove that a triangular matrix with non-zero entries on the leading diagonal is invertible.

21. Show that a triangular matrix may not be invertible. 3 Page 22. Show that the Wronskian W e2xcos x , e 2 x sin xProblems 0 . on Chapter 6

23. Find any errors in the following derivation. You must say what the error is and why. Let A and B be square matrices. Then 2 2 detABAB   det     2 detAB  det   

x 1 1 1    1x 1 1 24. Let A    . Find 1 1x 1    1 1 1 x  (i) determinant of the matrix A. (ii) the value of x for which the matrix is non-invertible.

25. An anti-symmetric matrix A is a square matrix such that AAT   . Let A be an n by n anti-symmetric matrix. (i) Prove that if n is odd then detA  0. (ii) Find a formula for det AAm T  . 4 Page Brief Solutions to Problems of Chapter 6 Problems on Chapter 6 2. k  2, 1 1 3. (a) 8 (b) 46 875 (c) 7776 (d)  6 4. 0 5. 40 9. 1 10. y11  x 11. 1 12. a5 b 5

13. (i) detAAAAA2  det 3   1, det 4 det 5  1, det 6    1

n/2  (ii) detAn    1

1 16 4  2 14. XXA ,  20 1 4  15. (ii) 1 in each case 17. The result is false. 19. No 23. Error in the last line. 3 24. (i) detA  x  1  3  x (ii) 1, 3

m1 detA  if n is even 25. (ii)   0 if n is odd 5 Page