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Calcul matriciel
Opérations sur les matrices : Addition et soustraction - Exercice 2
5 min
10
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{\color{red}\underline{COMPETENCE}\;:\;Calculer}
COMPETENCE
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Question 1
On considère les matrices suivantes :
A
=
(
2
1
4
1
0
7
)
A=\left(\begin{array}{ccc} {2} & {1} & {4} \\ {1} & {0} & {7} \end{array}\right)
A
=
(
2
1
1
0
4
7
)
;
B
=
(
1
0
3
2
5
6
)
B=\left(\begin{array}{ccc} {1} & {0} & {3} \\ {2} & {5} & {6} \end{array}\right)
B
=
(
1
2
0
5
3
6
)
et
C
=
(
0
−
1
2
1
−
5
−
2
)
C=\left(\begin{array}{ccc} {0} & {-1} & {2} \\ {1} & {-5} & {-2} \end{array}\right)
C
=
(
0
1
−
1
−
5
2
−
2
)
Calculer
A
+
B
A+B
A
+
B
Correction
A
+
B
=
(
2
1
4
1
0
7
)
+
(
1
0
3
2
5
6
)
A+B=\left(\begin{array}{ccc} {\red{2}} & {\blue{1}} & {\pink{4}} \\ {\green{1}} & {\orange{0}} & {7} \end{array}\right)+\left(\begin{array}{ccc} {\red{1}} & {\blue{0}} & {\pink{3}} \\ {\green{2}} & {\orange{5}} & {6} \end{array}\right)
A
+
B
=
(
2
1
1
0
4
7
)
+
(
1
2
0
5
3
6
)
A
+
B
=
(
2
+
1
1
+
0
4
+
3
1
+
2
0
+
5
7
+
6
)
A+B=\left(\begin{array}{ccc} {\red{2}+\red{1}} & {\blue{1}+\blue{0}} & {\pink{4}+\pink{3}} \\ {\green{1}+\green{2}} & {\orange{0}+\orange{5}} & {7+6} \end{array}\right)
A
+
B
=
(
2
+
1
1
+
2
1
+
0
0
+
5
4
+
3
7
+
6
)
A
+
B
=
(
3
1
7
3
5
13
)
A+B=\left(\begin{array}{ccc} {\red{3}} & {\blue{1}} & {\pink{7}} \\ {\green{3}} & {\orange{5}} & {13} \end{array}\right)
A
+
B
=
(
3
3
1
5
7
13
)
Question 2
Calculer
2
A
2A
2
A
Correction
Le produit d'une matrice
A
A
A
par un réel
k
k
k
, est la matrice
k
A
kA
k
A
, obtenue en multipliant chaque coefficient de
A
A
A
par
k
k
k
.
Nous savons que :
A
=
(
2
1
4
1
0
7
)
A=\left(\begin{array}{ccc} {2} & {1} & {4} \\ {1} & {0} & {7} \end{array}\right)
A
=
(
2
1
1
0
4
7
)
Ainsi :
2
A
=
(
2
×
2
2
×
1
2
×
4
2
×
1
2
×
0
2
×
7
)
\purple{2}A=\left(\begin{array}{ccc} {\purple{2}\times 2} & {\purple{2}\times 1} & {\purple{2}\times 4} \\ {\purple{2}\times 1} & {\purple{2}\times 0} & {\purple{2}\times 7} \end{array}\right)
2
A
=
(
2
×
2
2
×
1
2
×
1
2
×
0
2
×
4
2
×
7
)
2
A
=
(
4
2
8
2
0
14
)
2A=\left(\begin{array}{ccc} {4} & {2} & {8} \\ {2} & {0} & {14} \end{array}\right)
2
A
=
(
4
2
2
0
8
14
)
Question 3
Calculer
A
+
C
A+C
A
+
C
Correction
A
+
C
=
(
2
1
4
1
0
7
)
+
(
0
−
1
2
1
−
5
−
2
)
A+C=\left(\begin{array}{ccc} {\red{2}} & {\blue{1}} & {\pink{4}} \\ {\green{1}} & {\orange{0}} & {7} \end{array}\right)+\left(\begin{array}{ccc} {\red{0}} & {\blue{-1}} & {\pink{2}} \\ {\green{1}} & {\orange{-5}} & {-2} \end{array}\right)
A
+
C
=
(
2
1
1
0
4
7
)
+
(
0
1
−
1
−
5
2
−
2
)
A
+
C
=
(
2
+
0
1
+
(
−
1
)
4
+
2
1
+
1
0
+
(
−
5
)
7
+
(
−
2
)
)
A+C=\left(\begin{array}{ccc} {\red{2}+\red{0}} & {\blue{1}+\blue{\left(-1\right)}} & {\pink{4}+\pink{2}} \\ {\green{1}+\green{1}} & {\orange{0}+\orange{\left(-5\right)}} & {7+\left(-2\right)} \end{array}\right)
A
+
C
=
(
2
+
0
1
+
1
1
+
(
−
1
)
0
+
(
−
5
)
4
+
2
7
+
(
−
2
)
)
A
+
C
=
(
3
0
6
2
−
5
5
)
A+C=\left(\begin{array}{ccc} {\red{3}} & {\blue{0}} & {\pink{6}} \\ {\green{2}} & {\orange{-5}} & {5} \end{array}\right)
A
+
C
=
(
3
2
0
−
5
6
5
)