Exercices Matrices
Ruben Hillewaere
Inspiré du syllabus de L. Zandarin et autres ouvrages
ECAM
Décembre 2024
Contents
1 Linear Algebra & Matrices 2
1.1 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Solutions exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1
Chapter 1
Linear Algebra & Matrices
1.1 Exercices
1. Write the matrices representing the following linear transformations:
(a) the clockwise rotation of 90◦
(b) the symmetry around the origin
(c) the projection on the line with equation y = x
(d) the symmetry around the line with equation y = x
(e) the anti-clockwise rotation of 45◦
2. What linear transformations are represented by the following matrices?
−1 0
(a) A =
0 1
0 −1
(b) B =
1 0
(c) AB
(d) BA
(e) A−1
(f) B −1
Then, compute AB, BA, A−1 and B −1 to verify your answers.
3. Given the vectors
⃗ = (1, −2),
X Y⃗ = (2, 0), ⃗ = (3, −1)
Z
2
1.1. EXERCICES 3
Using matrix multiplication, compute:
⃗ and Y⃗
(a) the scalar product of X
⃗
(b) the length of Z
⃗ by the rotation of 30◦ (anti-clockwise)
(c) the image of X
4. Given the matrix
1 1 −1
A = −2 −1 1
3 2 −1
compute:
(a) the determinant |A|
(b) the adjoint matrix adj(A)
(c) the inverse matrix A−1
Verify your answer by computing A.A−1 or A−1 .A.
5. Use your results of the previous exercice to solve the linear system of equations
x + y − z = −3
−2x − y + z = 4
3x + 2y − z = −6
Verify your answer!
6. Verify whether the vectors
2 −1 −1
V1 = −1 , V2 = 4 , V3 = 11
0 7 21
are linearly dependent or independent. If they are linearly dependent, express V3 as
a linear combination of V1 and V2 . What does this mean geometrically?
7. Verify whether the vectors
−1 1 5
V1 = 4 , V2 = 3 , V3 = −6
11 10 −13
are linearly dependent or independent. If they are linearly dependent, express V3 as
a linear combination of V1 and V2 . What does this mean geometrically?
8. (a) Show that the vectors
1 1 −3
V1 = 1 , V2 = −1 , V3 = 2
2 1 −1
form a basis in R3 .
1.2. SOLUTIONS EXERCICES 4
(b) Compute the components of the vector
−10
3
−6
with respect to that basis.
1.2 Solutions exercices
0 1
1. (a)
−1 0
−1 0
(b)
0 −1
1/2 1/2
(c)
1/2 1/2
0 1
(d)
1 0
√ √
√ 2/2 −
√ 2/2
(e)
2/2 2/2
2. (a) Symmetry around the y -axis (or 2-axis)
(b) Rotation of 90◦ (anti-clockwise)
(c) The rotation followed by the symmetry
(d) The symmetry followed by the rotation
(e) The symmetry itself
(f) Rotation of -90◦ (that is clockwise)
0 1
(g) AB =
1 0
0 −1
(h) BA =
−1 0
−1 −1 0
(i) A =
0 1
−1 0 1
(j) B =
−1 0
3. (a) 2
1.2. SOLUTIONS EXERCICES 5
√
(b) 10
√
3
2 +√1
(c) 1
2 − 3
4. (a) 1
−1 −1 0
(b) adj(A) = 1 2 1
−1 1 1
(c) A−1 = adj(A)
5. (x, y , z, ) = (−1, −1, 1)
6. Dependent, and V3 = V1 + 3V2 . The vectors are lying in the same plane.
7. Dependent, and V3 = −3V1 + 2V2 . The vectors are lying in the same plane.
8. (a) |A| = −5
−2
(b) 1
3
