Linear Algebra Exercices
List 2: Elementary matrices, Rank and the Inverse Matrix
1. Let
A=
0 0 1
1 1 0
1 1 1
Compute
(Tr1r3)·A=
0 0 1
0 1 0
1 0 0
·A
Same question with A·(Tc1c2).
2. For the same matrixA as in previous exercice, computeP(−1)r3·A and Sr2+(−1)r1 ·Tr1r3 ·A
3. Compute the echelon reduced form of A=
1 2 1 −1
B =
1 1 1
2 1 0
Determine rank (A) and rank (B).
4. Let A∈ M3×3 be such that
Tr3−r1 ·P2r2·A =I3 Determine A.
Hint: Multiply the given equation by inverses of elementary matrices and “isolate” the unknown A.
5. A square matrix is said to be ortogonal if and only if AT =A−1
(a) Verify that
1 9
1 8 −4
4 −4 −7
8 1 4
is orthogonal.
(b) Prove that ifA and B are n×n orthogonal matrices, thenAB is an orthogonal matrix.
(c) Let
A =
cos(θ) 0 −sin(θ)
0 1 0
sin(θ) 0 cos(θ)
·
1 0 0
0 cos(φ) −sin(φ) 0 sin(φ) cos(φ)
Prove thatA is orthogonal.
6. Compute the inverse of the matrices A=
0 1 0
3 0 0
0 1 1
B =
1 −1 1
0 1 −1
1 3 −2
7. Using well known properties of the inverse matrix, compute (AT)−1, (A2)−1, (ATB)−1
where A and B are given in the previous exercice.
Hints or solutions:
Exercice 1
(Tr1r3)·A=
1 1 1
1 1 0
0 0 1
A·Tc1c2 =A Exercice 3
The echelon reduced form of A isI2. The echelon reduced form of B is 1 0 −1
0 1 2
We have rank(A) = rank(B) = 2.
Exercice 4
A=P1
2r2 ·Tr3+r1 ·I3 =
1 0 0
0 12 1
1 0 1
Exercice 5
(a) ConfirmA·AT =I. (b) Recall that (AB)T =BTAT. (c) Use (b).
Exercice 6 Obtain
A−1 =
0 1/3 0
1 0 0
−1 0 1
B−1 =
1 1 0
−1 −3 1
−1 −4 1
Exercice 7
Note that (AT)−1 = (A−1)T, (A2)−1 = (A−1)2, etc ... (revise properties of the inverse and transpose matrix in section 1.3 if necessary).
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. Complete the following matricial equalities:
(a) If A, B, C ∈ Mn×n are invertible, then ABC is invertible and (ABC)−1 =...
(b) If A∈ Mn×n is invertible, then AT is invertible and (3AT)−1 =...
4. Let A, B ∈ Mn×n be invertible matrices. Simplify (ABT)T(BAT)−1 . 5. Compute the inverse of the matrix
0 0 2
0 1 0
1 −1 0
6. Let
A =
1 −1 2
0 1 1
1 −1 1
and B =
−2 1 3
−1 1 1
1 0 −1
(a) Compute A·B and conclude about the invertibility of A.
(b) Solve the linear system
1 −1 2
0 1 1
1 −1 1
·
x y z
=
1 1 0
(Hint: the system can be quickly solved once you know the inverse of A)
7. 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)
Supplementary exercices:
Problems 2.38-2.58, Examples 3.19-3.21, Exercices 3.15-3.20; Linear Al- gebra, Lispchutz and Lipson, sixth edition.
