Linear Algebra Exercices
List 3: Applications of Gauss Method and Determinant of a Matrix
1. Let
A =
1 0 0
0 1 0
3 0 1
and B =
1 0 0
−1 1 1
2 0 1
Compute the inverses of A, B and AB.
2. Compute the inverses of the following upper triangular matrices
A2 =
1 1
0 1
, A3 =
1 1 1
0 1 1
0 0 1
, A4 =
1 1 1 1
0 1 1 1
0 0 1 1
0 0 0 1
and conjecture what would be the inverse of A5.
3. (Adapted from Linear Algebra, Lipschutz and Lipson). Consider the system
(S) :=
x1+x2−2x3+ 4x4 = 5 2x1+ 2x2−3x3+x4 = 3 3x1+ 3x2−4x3−2x4 = 1
(a) Write the system (S) as AX = B and set in echelon form the augmented matrix [A|B].
(b) Show that the system is possible, undetermined, with two degrees of freedom.
4. Study, according the values of the parameter ofk, the solvability of the system
x+ 2y+ 3z = 1 y+kz = 1 ky+k2z = 1 5. Consider the following matrices
A=
−1 13 1 23
B =
3 3 π
0 2 √
2
0 0 1
C=
2 3 4
5 4 3
1 2 1
D=
1 0 0
9 −1 2
1 2 −4
(a) Compute |A|,|B|,|C|, |D|, |CT|, |BC|, |B−1|.
(b) Which of these matrices are non-invertible?
6. (Geometric aspects of the determinant)
(a) Compute the area of the parallelogram defined by the vectors (1,1) and (−1,1).
(b) Compute the volume of the parallelepiped defined by the vectors (1,2,2), (2,1,2) and (2,2,1).
(c) Given three vectorsu1, u2 and u3 inR3, suppose the determinant of a 3×3 matrix whose rows equalu1,u2 andu3 is null. What can we say about the parallelelepiped defined by this three vectors?
7. Use Laplace’s theorem to compute
0 2 0 1
5 π−1 0 e−1
π e 1 √
3
0 2 0 4
8. Use Cramer’s rule to solve the following systems:
(x+ 2y= 4
−x+y =−1
x+y+z = 3
−x+y+z = 1
−x−y+z =−1
9. (adapted from Linear Algebra, Lipschutz and Lipson).
Let
C=
2 3 −4
0 −4 2
1 −1 5
Fill the gaps
Adj(C) =
−18 −11 −10
x 14 y
4 z −8
and compute C−1. Test and exam exercices:
1. Set the matrix C =
1 −2 1 0
0 2 2 −1
in echelon reduced form.
2. Compute the inverse of A=
0 0 1
0 2 −1
1 0 3
3. Consider the system
x−2y = 1 x−y+az = 2 ay+ 9z = 3
where a is a real parameter. Study the existence of solutions to the system according to the values of a. In the undetermined case, indi- cate the number of free variables of the system.
4. Consider the system
x+αy+βz = 1 α(β−1)y =α x+αy+z =β2 where α, β are real parameters.
Discuss the existence/non existence of solutions to the system according to the values of αand β. In the existence cases, you should refer if the system is determined or indetermined, indicating, in the last case, the number of degrees of freedom.
5. (a) Let
A=
0 −2 1
1 π e
0 3 1
Use Laplace’s theorem to verify that|A|= 5 and compute|2A−1AT| using properties of the determinant.
(b) LetA∈ Mn×n be such that
In =Ek·...·E1·A (k ∈N)
where eachElis an elementary matrix of typeSri+αrj orTrirj (i.e.
performing the substitution of rowri byri+αrj or interchanging row ri with row rj). Show that |A|= 1 or |A|=−1.
6. Consider the system
x+y+z = 1
x+ 2y+ (1−α)z = 3 y+α2z =α+ 2
where α is a real pa- rameter.
(a) Classify the system by the number of solutions, according to para- meter α.
(b) Set α= 2. Use Cramer’s rule to discover y.
7. Let A be a 4×4 matrix such that |A| = 13. Let B = [bij] be an upper triangular 4×4 matrix such that
bii = 1
i i= 1, ...,4 Compute |2A−1B|.
8. Let
A=
−1 1 0
−2 1 0
0 0 3
(i) Verify A is invertible and explicit its inverse.
(ii) Let B be a 3×3 invertible matrix such that
B−1 =
1 1 1
0 1 1
0 0 1
Compute (A−1BT)−1. (Hint: use well known properties of matrix in- version)
9. Let
L=
3 1 2
0 3 7
1 0 1
Consider the system
L·
x y z
=
a b 0
Show that x= (3a−b)/|L| (hint: use Cramer’s rule).
Supplementary exercices:
Linear Algebra, Lipschutz and Lipson, 6th edition, Mc Graw Hill.
Applications of Gaus method: 3.15, 3.17, 3.18, 3.20, 3.21, 3.31, 3.32, 3.33 Determinants: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.9, 8.11, 8.12
