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Calcul matriciel
Opérations sur les matrices : Addition et soustraction - Exercice 4
12 min
20
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{\color{red}\underline{COMPETENCE}\;:\;Calculer}
COMPETENCE
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Question 1
On considère les matrices suivantes :
A
=
(
−
2
3
4
1
)
A=\left(\begin{array}{cc} {-2} & {3} \\ {4} & {1} \end{array}\right)
A
=
(
−
2
4
3
1
)
et
B
=
(
5
6
2
−
5
)
B=\left(\begin{array}{cc} {5} & {6} \\ {2} & {-5} \end{array}\right)
B
=
(
5
2
6
−
5
)
Calculer :
A
+
B
A+B
A
+
B
Correction
A
+
B
=
(
−
2
3
4
1
)
+
(
5
6
2
−
5
)
A+B=\left(\begin{array}{cc} {\red{-2}} & {\blue{3}} \\ {\green{4}} & {\pink{1}} \end{array}\right)+\left(\begin{array}{cc} {\red{5}} & {\blue{6}} \\ {\green{2}} & {\pink{-5}} \end{array}\right)
A
+
B
=
(
−
2
4
3
1
)
+
(
5
2
6
−
5
)
A
+
B
=
(
−
2
+
5
3
+
6
4
+
2
1
+
(
−
5
)
)
A+B=\left(\begin{array}{cc} {\red{-2}+\red{5}} & {\blue{3}+\blue{6}} \\ {\green{4}+\green{2}} & {\pink{1}+\left(\pink{-5}\right)} \end{array}\right)
A
+
B
=
(
−
2
+
5
4
+
2
3
+
6
1
+
(
−
5
)
)
A
+
B
=
(
3
9
6
−
4
)
A+B=\left(\begin{array}{cc} {3} & {9} \\ {6} & {-4} \end{array}\right)
A
+
B
=
(
3
6
9
−
4
)
Question 2
Calculer :
A
−
B
A-B
A
−
B
Correction
A
−
B
=
(
−
2
3
4
1
)
−
(
5
6
2
−
5
)
A-B=\left(\begin{array}{cc} {\red{-2}} & {\blue{3}} \\ {\green{4}} & {\pink{1}} \end{array}\right)-\left(\begin{array}{cc} {\red{5}} & {\blue{6}} \\ {\green{2}} & {\pink{-5}} \end{array}\right)
A
−
B
=
(
−
2
4
3
1
)
−
(
5
2
6
−
5
)
A
−
B
=
(
−
2
−
5
3
−
6
4
−
2
1
−
(
−
5
)
)
A-B=\left(\begin{array}{cc} {\red{-2}-\red{5}} & {\blue{3}-\blue{6}} \\ {\green{4}-\green{2}} & {\pink{1}-\left(\pink{-5}\right)} \end{array}\right)
A
−
B
=
(
−
2
−
5
4
−
2
3
−
6
1
−
(
−
5
)
)
A
−
B
=
(
−
7
−
3
2
6
)
A-B=\left(\begin{array}{cc} {-7} & {-3} \\ {2} & {6} \end{array}\right)
A
−
B
=
(
−
7
2
−
3
6
)
Question 3
Calculer :
3
A
3A
3
A
Correction
3
A
=
3
×
(
−
2
3
4
1
)
\purple{3}A=\purple{3}\times\left(\begin{array}{cc} {-2} & {3} \\ {4} & {1} \end{array}\right)
3
A
=
3
×
(
−
2
4
3
1
)
3
A
=
(
−
2
×
3
3
×
3
4
×
3
1
×
3
)
\purple{3}A=\left(\begin{array}{cc} {-2\times\purple{3}} & {3\times\purple{3}} \\ {4\times\purple{3}} & {1\times\purple{3}} \end{array}\right)
3
A
=
(
−
2
×
3
4
×
3
3
×
3
1
×
3
)
3
A
=
(
−
6
9
12
3
)
\purple{3}A=\left(\begin{array}{cc} {-6} & {9} \\ {12} & {3} \end{array}\right)
3
A
=
(
−
6
12
9
3
)
Question 4
Calculer :
4
B
4B
4
B
Correction
4
B
=
4
×
(
5
6
2
−
5
)
{\color{blue}{4}}B={\color{blue}{4}}\times\left(\begin{array}{cc} {5} & {6} \\ {2} & {-5} \end{array}\right)
4
B
=
4
×
(
5
2
6
−
5
)
4
B
=
(
5
×
4
6
×
4
2
×
4
−
5
×
4
)
{\color{blue}{4}}B=\left(\begin{array}{cc} {5\times\color{blue}{4}} & {6\times\color{blue}{4}} \\ {2\times\color{blue}{4}} & {-5\times\color{blue}{4}} \end{array}\right)
4
B
=
(
5
×
4
2
×
4
6
×
4
−
5
×
4
)
4
B
=
(
20
24
8
−
20
)
{\color{blue}{4}}B=\left(\begin{array}{cc} {20} & {24} \\ {8} & {-20} \end{array}\right)
4
B
=
(
20
8
24
−
20
)
Question 5
Calculer :
3
A
−
4
B
3A-4B
3
A
−
4
B
Correction
D'après les questions précédentes, nous savons que :
3
A
=
(
−
6
9
12
3
)
3A=\left(\begin{array}{cc} {-6} & {9} \\ {12} & {3} \end{array}\right)
3
A
=
(
−
6
12
9
3
)
et
4
B
=
(
20
24
8
−
20
)
4B=\left(\begin{array}{cc} {20} & {24} \\ {8} & {-20} \end{array}\right)
4
B
=
(
20
8
24
−
20
)
On a :
3
A
−
4
B
=
(
−
6
9
12
3
)
−
(
20
24
8
−
20
)
3A-4B=\left(\begin{array}{cc} {\red{-6}} & {\blue{9}} \\ {\green{12}} & {\pink{3}} \end{array}\right)-\left(\begin{array}{cc} {\red{20}} & {\blue{24}} \\ {\green{8}} & {\pink{-20}} \end{array}\right)
3
A
−
4
B
=
(
−
6
12
9
3
)
−
(
20
8
24
−
20
)
3
A
−
4
B
=
(
−
6
−
20
9
−
24
12
−
8
3
−
(
−
20
)
)
3A-4B=\left(\begin{array}{cc} {\red{-6}-\red{20}} & {\blue{9}-\blue{24}} \\ {\green{12}-\green{8}} & {\pink{3}-\left(\pink{-20}\right)} \end{array}\right)
3
A
−
4
B
=
(
−
6
−
20
12
−
8
9
−
24
3
−
(
−
20
)
)
3
A
−
4
B
=
(
−
26
−
15
4
23
)
3A-4B=\left(\begin{array}{cc} {-26} & {-15} \\ {4} & {23} \end{array}\right)
3
A
−
4
B
=
(
−
26
4
−
15
23
)