Algebra Linear e Geometria AnalıticaAlgebra Linear e Geometria AnalıticaDepartamento de Matematica FCT-UNL

1st semester - 2014/2015

Exercises: “Matrices”

1. Consider the matrix A =

[11 1

2 −2 34 1 1 0

].

(a) What is the size of the matrix A?

(b) What are the values of the elements a12 e a21?

(c) For which pairs (i, j) we have aij = 1?

(d) Indicate the 2nd column and the 1st row of A.

2. Find the solution to each matricial equations with unknowns a, b, c and d:

(a)

[1 23 a+ 2

]=

[1 23 5

]; (b)

[a− b b+ a3d+ c 2d− c

]=

[8 17 6

].

3. Consider the matrices A =

3 0−1 21 1

, B =

2 1−3 14 0

, C =

[1 03 −1

]e D =[

1 1−3 3

]. Compute, if it is possible:

(a) A+ 2B; (b) 3B +D; (c) 4D − 3C.

4. Indicate the matrix A = [aij ]i,j∈{1,2,3} defined by:

(a) aij = i+ j;

(b) aij =

0, i = j−1, i > j2, i < j.

5. Find a, b and c real numbers such that:

1 2 13 −1 20 1 1

2−1a

=

2bc

.

6. Consider the matrices M3×3(R),

A =

1 0 00 1 00 0 1

e B =

1 1 11 1 11 1 1

.Determine a matrix X ∈M3×3(R), that satisfies the equation

X +A = 2(X −B).

7. Compute, if it is possible, the products AB e BA, when:

(a) A =[

1 2 3]

e B =

0−10

; (b) A =[

1 2 3]

e B =

−1 00 11 0

;

(c) A =

[1 00 0

]e B =

[0 10 0

];

(d) A =

[1 00 1

]e B =

[a bc d

];

(e) A =

[0 00 0

]e B =

[a bc d

].

8. For each pair of matrices, compute, if it is possible, the matrices A2, B2, AB e BA.

(a) A =

[1 0 32 1 0

]e B =

2 1−1 24 1

;

(b) A =

[1 11 1

]e B =

[0 10 0

];

(c) A =[

3]

e B =

123

;

(d) A =

[1 0 11 −1 2

]e

B =

[−1 0 20 1 0

].

9. Using the matrices A e B of the exercise 8(a), compute: AB, (AB)T , BTAT e ATBT .

10. Consider the following matrices:

A =

−1 2 32 0 13 1 2

, B =

1 2 2−2 0 0−2 0 0

, C =

1 11 11 1

D =

0 0 00 0 00 0 0

, E =

0 2 1−2 0 −3−1 3 0

.Indicate which matrices are symmetric or skew-symmetric.

11. Show that the null matrix of order n is the only matrix that is simultaneously sym-metric and skew-symmetric.

12. Using the matrices of the exercise 3, compute if it is possible

(a) AD +BD;

(b) BD +A;

(c) (A+B)D;

(d) DT (AT +BT );

(e) (C2)T ;

(f) (CT )2.

13. Using the matrices A e B of the exercise 8a), indicate (AB)12, the first row of AB andthe third column of BA.

14. Justify if the following statements are true or false:

(a) If A and B are square matrices with the same order, then

(A+B)(A−B) = A2 −B2.

(b) If A and B are square matrices with the same order, then

(A+B)2 = A2 + 2AB +B2.

(c) If A and B are square matrices with the same order, then

(AB)n = AnBn

for each n ∈ N.

15. Find an expression to An, n ∈ N, where A is the matrix:

2

(a)

[1 00 1

];

(b)

[0 00 0

];

(c)

[0 10 0

];

(d)

[1 00 0

];

(e)

[0 −11 0

];

(f)

[cos(α) sin(α)− sin(α) cos(α)

].

16. Consider the matrices

A =

1 −1 11 0 12 −1 0

B =

0 1 0 3 00 0 0 1 10 0 0 0 0

C =

2 −1 0 20 1 1 21 −1 2 −10 0 3 1

D =

1 1 −10 2 30 5 0

E =

0 0 0 00 2 4 00 0 8 30 0 0 0

F =

0 1 0 30 0 1 20 0 0 0

G =

2 −1 00 0 30 0 6

H =

1 0 00 1 00 0 1

I =

0 0 00 0 00 0 0

(a) Determine whether the matrix is in the row-echelon form and in the reduced

echelon-form

(b) Find the rank of each of the matrices.

(c) Indicate the reduced echelon-form to each matrix that are not in this form.

17. Indicate if the following matrices are in the reduced echelon-form:

(a)

[1 1 10 0 2

]e

[0 1 01 −1 0

]; (b)

[1 2 3−2 −4 −5

]e

[1 2 11 2 0

].

18. Compute the rank of the following matrices:

A =

1 −1 11 1 13 −1 35 −3 5

B =

1 −1 1 31 1 1 03 −1 3 45 −3 5 10

C =

1 −1 1 31 1 1 α3 −1 3 45 −3 5 10

.19. Find the rank of the following matrices for each α, β ∈ R.

Aα =

1 0 −1 11 1 0 1α 1 −1 2

Bα =

1 −1 0 11 1 0 −1α 1 1 00 1 α 1

Cα,β =

0 0 α0 β 23 0 1

.

20. Let A ∈M3×3(R) be an invertible matrix, such taht

A−1 =

1 1 20 1 34 2 1

.(a) Find a matrix B such that AB =

1 20 14 1

.(b) Find a matrix C such that AC = A+2I3 and justify that the matrix C is unique.

21. Determine whether the matrix is invertible; if so find the inverse.

3

(a)

1 0 00 2 00 0 3

;

(b)

0 0 10 2 03 0 0

;

(c)

a 0 00 b 00 0 c

;

(d)

0 0 c0 b 0a 0 0

;

(e)

a 0 01 a 00 1 a

.

22. Determine whether the matrix is invertible.

(a)

0 1 01 0 00 0 1

; (b)

1 1 12 2 23 3 3

; (c)

1 1 1−1 −3 −22 4 3

.23. Consider the matrices

A =

[1 0−3 1

]B =

1 0 00 1 00 0 3

C =

0 0 0 10 1 0 00 0 1 01 0 0 0

D =

1 0 − 1

7 00 1 0 00 0 1 00 0 0 1

E =

1 0 00 1 00 0 0

F =

1 0 0 00 1 0 00 0 1 00 12 0 1

G =

−1 0 00 1 00 0 1

(a) Determine whether the matrix is an elementary matrix and if so indicate its type(I, II or III).I

(b) By inspection indicate the product of:

A

[2 7 19 9 2

], C

1 α 34 5 βγ 0 11 7 0

, D

78

18 1

6 120 0

0 78 0

0 0 3

, B

1 50 10 1

3

.

(c) Indicate the inverse of each of the elementary matrices of (a).

24. Consider the matrix A =

1 0 −20 1 00 0 2

.

(a) Determine elementary matrices E1 e E2 such that E1E2A = I.

(b) Write A−1 as a product of two elementary matrices.

(c) Write A as a product of two elementary matrices.

25. Write A and A−1 as a product of elementary matrices.

(a)

2 1 11 2 11 1 2

; (b)

1 1 01 1 10 1 1

.

4

26. Consider the matrix

A =

0 1 7 81 3 3 8−2 −5 1 −8

.Find elementary matrices E,F and G and R a row-echelon form matrix em formasuch that:

A = EFGR.

27. Compute, if exists, the inverse of each of the following matrices:

(a)

3 4 −11 0 32 5 −4

;

(b)

−1 3 −42 4 1−4 2 −9

;

(c)

1 0 10 1 11 1 0

;

(d)

1 2 02 1 20 2 1

;

(e)

2 0 00 4 30 1 1

;

(f)

1 2 30 2 30 0 3

.28. (a) Compute the set of values α ∈ R such that the matrix 1 2 0

1 4 22 4 5 + α

is invertible.

(b) Compute the set of values α ∈ R and the set of values β ∈ R such that the matrix 1 2 11 α+ 3 22 4 β

is invertible.

29. Justify if each the following statements are true or false:

(a) Each invertible matrix can be written as a product of elementary matrices. ele-mentares.

(b) If A is a singular square matrix, then the system AX = B has infenitely manysolutions.

(c) If A is a singular square matrix, then its reduced echelon form has at least a nullrow.

(d) If A can be written as a product of elementary matrices, then the linear homo-geneous system AX = 0 has only the null solution.

(e) If A is a singular square matrix and if B was obtained from A interchanging tworows, then B is also a singular matrix.

5

Algebra Linear e Geometria AnalıticaAlgebra Linear e Geometria AnalıticaDepartamento de Matematica FCT-UNL

1st semester - 2014/2015

Exercises: “Systems of Linear Equations”

1. Which of the following are linear equations in x, y and z?

(a) x+ πy + 3√

2z = e;

(b) x2 + xy + z = 2;

(c) sin y + 2z = −x;

(d) ax+ 6y − 3z = 9, com a ∈ R.

2. Which geometric figure in space R3 can a system of linear equations in 3 unknownsrepresent? ?

3. Consider the linear system in x1, x2, x3, x4, over R,x1 + x2 + 2x3 − x4 = −12x1 + 2x2 − 2x3 + 2x4 = 4

6x3 − 4x4 = −6.

(a) For the matricial form of the system AX = B indicate the coefficient matrix, theunknowns matrix, the matrix of constant terms and the augmented matrix.

(b) What is the matrix form of the system?

(c) Justify in two distinct ways that (x1, x2, x3, x4) = (−1, 1, 1, 3) is a solution forthe system.

4. Find, geometrically and analytically, the solution of the following linear systems withthe unknowns x1 and x2 in R:

(a)

{2x1 + 2x2 = 03x1 − x2 = 0.

(b)

{2x1 + 2x2 = 2−x1 − x2 = −1.

5. Find the set of solution of the following systems and indicate if it is consistent (withexactly one solution or with infinitely many solutions) or inconsistent:

(a)

x+ z = 1x+ y = 3y + z = 2.

(b)

x+ y − 3z = −12x+ y − 2z = 1x+ y + z = 3x+ 2y − 3z = 1.

(c)

x1 + x2 − x3 = 02x1 + x2 = 1x1 − x3 = 12x1 + x2 − 2x3 = 1.

(d)

x+ 2y − 2z − w = 12x+ y − z + 4w = −1−3x+ 3z + 9w = 0.

(e)

2x1 + x2 = 1−x1 + 3x2 + x3 = 2x1 + 4x2 + x3 = 3.

(f)

−4x+ 2y + 2z = 87x+ y − 8z = −5−4x+ 3y + z = 10.

(g)

{x1 + 2x2 − x3 = −12x1 + 4x2 + 2x3 = 3.

(h)

x+ 2y − 2z − w = 12x+ y − z + 4w = −1−3x+ 3z + 9w = 0.

(i)

x+ 7y + 5z + 3w + 2u = 04y + 2z + 2w = 12x− 2y + 4z + u = −13x− y + 7z + w + 3u = 0.

6. By inspection (without calculations)indicate whether the systems have multiple solu-tions:

(a)

{a11x1 + a12x2 + a13x3 = 0a21x1 + a22x2 + a23x3 = 0.

(b)

{5x1 + 3x2 = 215x1 + 1

3x2 = 12 .

(c)

x1 − x2 + x3 − x4 = 0x1 − x2 + 3x3 − 2x4 = 0x1 + x2 + x3 + x4 = 0.

7. Find for what values of α the linear system with variables x1, x2, x3, with real constantshas non-null solutions:

x1 − αx2 + x3 = 0x1 − x2 + αx3 = 0x1 + αx2 + x3 = 0.

8. Consider the following system of linear equations with variables x, y, z and real cons-tants:

ax+ y + cz = 1x+ by + z = 1cx+ y + az = 1.

(a) For a = 0, b = 1 and c = 1 find the solutions of the system.

(b) For a = b = c = 1, find the rank of the coefficient matrix and of the augmentedmatrix of the system.

(c) Show that for all b ∈ R if a = c the system has many infinitely solutions.

9. Show that is invertible the matrix

A =

−3 2 −12 0 −2−1 1 1

Use A−1 to compute the solution of the linear system with unknowns x, y, z, and realconstants

−3x+ 2y − z = α2x− 2z = β−x+ y + z = γ

, α, β, γ ∈ R.

10. (a) Find for what values of α the linear system is consistent.x− y + z = 3x+ y + z = α3x− y + 3z = 45x− 3y + 5z = 10.

(b) For the value of α that you have find in (a), compute the rank of the followingmatrices:

A =

1 −1 11 1 13 −1 35 −3 5

B =

1 −1 1 31 1 1 03 −1 3 45 −3 5 10

C =

1 −1 1 31 1 1 α3 −1 3 45 −3 5 10

11. Consider the following system of linear equations with variables x, y, z and real cons-

tants.

7

(a)

x+ y + z = 1x+ y + (b+ 1)z = 3x+ y + (a− 1)z = a− 1

(b)

−x+ ay + bz = bx− y + z = 2x− ay + z = 1

(c)

x− by − az = −ax− 2y + 2z = 3x− by + 2z = 2

.

(d)

x+ y − z = 1−x− ay + 2z = −1−x− y + (a+ 1)z = b− 2

.

Find for what real values of a and b the system is inconsistent, consistent with one orinfinitely many solutions.

12. For α, β ∈ R, Consider the following system of linear equations with variables x, y, zand w and real constants:

x+ z + 2w = 0x+ y + z + (α+ 2)w = 02x+ y + (α+ 2)z + (α+ 4)w = 04x+ βy + 4z + 8w = β.

(a) Find for what values of α and β the system is inconsistent, consistent with oneor infinitely many solutions.

(b) For α = β = 0 indicate the set of the solutions of the system.

8