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Matrices sheet
1) consider the matrices
3 0
๐ด = [โ1 2] ,
1 1
6 1
๐ธ = [โ1 1
4 1
4 โ1
๐ต=[
],
0 2
1
๐ถ=[
3
4 2
],
1 5
1
๐ท = [โ1
3
5 2
0 1]
2 4
3
2]
3
Compute the following (If possible).
2)
โ3(๐ท + 2๐ธ)
(4๐ธ โ 2๐ท)
(2๐ด๐ + ๐ถ)
(๐ท๐ + ๐ธ ๐ )
1
1
( ๐ถ ๐ โ ๐ด)
2
4
(๐ต โ ๐ต๐ )
(2๐ธ ๐ โ 3๐ท๐ )
(๐ด๐ต)
(๐ด๐ต)C
๐ด(๐ต๐ถ)
(๐ถ ๐ ๐ต)๐ด๐
(๐ต๐ด๐ โ 2๐ถ)๐
๐ต๐ (๐ถ๐ถ ๐ โ ๐ด๐ ๐ด)๐
๐ท๐ ๐ธ ๐ โ (๐ธ๐ท)๐
(โ๐ด๐ถ)๐ + 5๐ท๐
let ๐ฆ = [๐ฆ1
๐11
๐ฆ2 โฆ โฆ โฆ ๐ฆ๐ ] , ๐ด = [ ๐21
โฎ
๐๐1
๐12
๐22
โฎ
๐๐2
โฏ ๐1๐
โฏ ๐2๐
]
โฎ
๐๐๐
show that the product ๐ฆ๐ด can be expressed as a linear combination of the row
matrices of A with the scalar coefficients coming from y
3)
In each part, find matrices A, x, and b that express the given system of linear
equations as a single matrix equation AX=b
a)
2๐ฅ1 โ 3๐ฅ2 + 5๐ฅ3 = 7
9๐ฅ1 โ ๐ฅ2 + ๐ฅ3 = โ1
๐ฅ1 + 5๐ฅ2 + 4๐ฅ3 = 0
1
Matrices sheet
b)
4๐ฅ1
โ 3๐ฅ3 + ๐ฅ4 = 1
5๐ฅ1 + ๐ฅ2
โ 8๐ฅ4 = 3
2๐ฅ1 โ 5๐ฅ2 + 9๐ฅ3 โ ๐ฅ4 = 0
3๐ฅ2 โ ๐ฅ3 + 7๐ฅ4 = 2
4)
Let ๐ด = [
3 1
] ,
5 2
๐ต=[
2 โ3
],
4 4
๐ถ=[
6
4
]
โ2 โ1
verify that
๐) (๐ดโ1 )โ1 = ๐ด
๐) (๐ต๐ )โ1 = (๐ตโ1 )๐
๐) (๐ด๐ต)โ1 = ๐ตโ1 ๐ดโ1
5)
๐) (๐ด๐ต๐ถ)โ1 = ๐ถ โ1 ๐ตโ1 ๐ดโ1
Use Matrix row operations to solve the following systems, also find๐ดโ1 of their
coefficient matrices .
a)
๐ฅ1 + 2๐ฅ2 + 3๐ฅ3 = 5
2 ๐ฅ1 + 5๐ฅ2 + 3๐ฅ3 = 3
๐ฅ1
+8๐ฅ3 = 17
b)
5 ๐ฅ1 + 3๐ฅ2 + 2๐ฅ3 = 4
3 ๐ฅ1 + 3๐ฅ2 + 2๐ฅ3 = 2
๐ฅ2 + ๐ฅ3 = 5
6)
Find all values of a, b, and c for which A is symmetric
2 ๐ โ 2๐ + 2๐
๐ด = [3
5
0
โ2
7)
2๐ + ๐ + ๐
๐+๐ ]
7
Find all values of a and b for which A and B are both not invertible.
๐ด=[
๐+๐โ1 0
]
0
3
5
๐ต=[
0
0
]
2๐ โ 3๐ โ 7
2
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