MATH+4031+HW+2 - Practice Computational and Proof exercises for determinants of matrices PDF

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Practice Computational and Proof exercises for determinants of matrices...

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MATH 4031 - HOMEWORK 2

October 01, 2018

Recall: • If E is an elementary matrix then   if E is of type 1 (interchange 2 rows), −1 detA = α if E is of type 2 (multiply a row by a nonzero constant α),   1 if E is of type 3 (add a multiple of one row to another row).

• An n × n matrix A is singular if and only if detA = 0. • For any n × n matrix A, detAT = detA.

• For any n × n matrices A and B, det(AB) = (detA)(detB). • Let A = (aij )n×n , Aij = (−1)i+j detMij be the cofactor of aij . Then ( detA if i = j , ai1 Aj1 + ai2 Aj2 + . . . + ain Ajn = 0 if i 6= j . • Denote 

A11  A21  adjA =  ..  . An1

then if detA 6= 0

A12 A22 ... An2

T  . . . A1n A11  A12 . . . A2n    =  .. ..  ..  . . .  . . . Ann A1n A−1 =

1

1 adjA. detA

A21 A22 .. . A2n

 . . . An1 . . . An2   ..  .. . .  . . . Ann

Problem 1. Let A be an n × n matrix. Prove that 1 . a, detA−1 = detA n b, detA = (detA)n . c, det(αA) = αn detA. Problem 2. Let A and B be n × n matrices. Prove that AB is nonsingular if and only if A and B are BOTH nonsingular. Problem 3. Let A and B be n × n matrices. Prove that if AB = I then BA = I . (Note that in the definition of the nonsingular matrix, a matrix A is nonsingular if and only if AB = BA = I for some matrix B. This problem tell us that AB = I is enough to conclude that A is nonsingular.) Problem 4. Let A be an n × n nonsingular matrix. Recall that A−1 = a, Show that  n 1 det (A−1 ) = det(adjA). detA b, Show that

1 adjA. detA

det(adjA) = (detA)n−1 .

Problem 5. Show that if A is nonsingular then adjA is also nonsingular and (adjA)−1 = [det(A−1 )]A = adj(A−1 ). Problem 6. Show that if A is singular then adjA is also singular. Problem 7. Show that if detA = 1 then adj(adjA) = A. Problem 8. Let A be a nonsingular n × n matrix. a, Prove that A = detA(adjA)−1 . b, Prove that A=

1 adj(adjA). (detA)n−2

Problem 9. a, Let A, B be square matrices. Prove that   A O = (det A)(det B). det O B b, Let A1 , A2 , . . . , Ak be square matrices. Prove that   A1 O . . . O det  O A2 . . . O  = (det A1 )(det A2 ) . . . (det Ak ). O O . . . Ak c, Let A1 , A2 , . . . , Ak be square matrices. Prove that the  A1 O A =  O A2 O O

matrix

 ... O ... O  . . . Ak

is nonsingular if and only if ALL of the matrices in the diagonal A1 , A2 , . . . , Ak are nonsingular. Problem 9. Compute the LU factorization of each of the following matrices       1 0 1 −2 1 2 1 1 1  3 5 6 ,  4 1 −2  ,  3 3 4  . 2 2 3 −6 −3 4 −2 2 7

2

Problem 10. Evaluate the following determinants. Write your answer as polynomials in x and λ.   2−x 3 4  1−λ 0 3   1 1−x 0 ,  2 −λ 0 . 0

2

0

−x

2

1−λ

Problem 11. Find λ for which the following determinants will be 0.     1−λ 3 2−λ 3 , . 1 1−λ 2 3−λ Problem 12. Compute the following determinants   3  0 0  1

0 2 1 0

1 0 2 3

 2 1 , 0 1

  2  0 3   1 5

0 1 1 1 2

0 0 0 4 1

0 2 0 2 0

 3 0 1 . 0 1

Problem 13. Let A and B be n × n matrices. Which of the following equations are true? det (A + B) = det A + det B,

det (AB ) = (det A)(det B),

det (AB) = det (BA).

Problem 14. Let A and B be nonsingular 3 × 3 matrices. Given that det A = 2, det B = 3. Compute det (AT B),

det (A−1 B),

det (A2 B 3 ),

det (B −1 AB),

det (2A),

det (3B −1 ),

det (AB)T .

Problem 15. Suppose that A = (aij )n×n is a matrix with property that A−1 = AT . Prove that aij =

Aij . det A

Problem 16. Show that if detA = 0 then A adjA = O. Problem 17. Compute detA and then find A given that  1 adjA = 3 2 Problem 18. Use   2x1 + 3x2 + x3 x1 + x2 + x3   3x1 + 4x2 + 2x3   x1 + 3x2 + x3 2x 1 − 2x 2 + x 3   x1 − 5x2 + x3

0 3 2

 1 4 , 3

 1 adjA =  0 1

2 4 2

 1 3 , 2

 3 0  adjA =  0 0

the Cramer’s rule to solve the following linear systems   = 1, = 4, x1 − x2 + 2x3 = 3, 2x1 + 3x2 − x3 = 1,   = 4. 7x1 + 3x2 + 4x3 = 7.   = 3, = 0, x1 + 2x2 + x3 = 2, 2x1 − 3x2 + x3 = 1,   = 5. 3x1 + x2 + 2x3 = 0.

3

0 1 0 1

0 2 4 2

 0 1 . 3 2

Hints: Problem 1. a, Use det(AB ) = (detA)(detB) and the fact that AA−1 = I . b, Use det(AB) = (detA)(detB) and induction. c, Let Ei be the elementary matrix of type 2 corresponding to multiply the row i by α. Note that αA = E1 E2 . . . En A. Problem 2. Note that: AB is singular if and only if det(AB) = 0. Problem 3. AB = I implies that (detA)(detB) = 1, so detA 6= 0 and A is nonsingular. Thus, there exists the inverse A−1 of A. Show that it must be B (that is A−1 = B). Problem 4. a, Apply Problem 1c with α =

1 detA

we have

det (A−1 ) = det



  n 1 1 adj A = det (adj A). det A det A

b, Use the above equation and the fact that det (A−1 ) =

1 . det A

Problem 5. 1 adjA. To prove the first equation, take the inverse matrices in both sides of the equation A−1 = detA 1 adjA. To prove the second equation, replace A by A−1 in both sides of the equation A−1 = detA Problem 6. Use problem 4b. Problem 7. The equation A−1 =

1 adjA detA

implies that if detA = 1 then A−1 = adjA. Use this twice!

Problem 8. 1 adjA. a, Take the inverse in bothsides of the equation A−1 = detA b, Use part a, and replace A by adjA in bothsides of the equation A−1 = det(adjA) = (detA)n−1 .

4

1 adjA detA

with the notice that...