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sheet 2 Autosaved .pdf


Original filename: sheet-2-Autosaved.pdf
Title: Matrices sheet
Author: mufic

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Matrices sheet
1) consider the matrices
3 0
๐ด = [โˆ’1 2] ,
1 1
6 1
๐ธ = [โˆ’1 1
4 1

4 โˆ’1
๐ต=[
],
0 2

1
๐ถ=[
3

4 2
],
1 5

1
๐ท = [โˆ’1
3

5 2
0 1]
2 4

3
2]
3

Compute the following (If possible).

2)

โˆ’3(๐ท + 2๐ธ)

(4๐ธ โˆ’ 2๐ท)

(2๐ด๐‘‡ + ๐ถ)

(๐ท๐‘‡ + ๐ธ ๐‘‡ )

1
1
( ๐ถ ๐‘‡ โˆ’ ๐ด)
2
4

(๐ต โˆ’ ๐ต๐‘‡ )

(2๐ธ ๐‘‡ โˆ’ 3๐ท๐‘‡ )

(๐ด๐ต)

(๐ด๐ต)C

๐ด(๐ต๐ถ)

(๐ถ ๐‘‡ ๐ต)๐ด๐‘‡

(๐ต๐ด๐‘‡ โˆ’ 2๐ถ)๐‘‡

๐ต๐‘‡ (๐ถ๐ถ ๐‘‡ โˆ’ ๐ด๐‘‡ ๐ด)๐‘‡

๐ท๐‘‡ ๐ธ ๐‘‡ โˆ’ (๐ธ๐ท)๐‘‡

(โˆ’๐ด๐ถ)๐‘‡ + 5๐ท๐‘‡

let ๐‘ฆ = [๐‘ฆ1

๐‘Ž11
๐‘ฆ2 โ€ฆ โ€ฆ โ€ฆ ๐‘ฆ๐‘š ] , ๐ด = [ ๐‘Ž21
โ‹ฎ
๐‘Ž๐‘š1

๐‘Ž12
๐‘Ž22
โ‹ฎ
๐‘Ž๐‘š2

โ‹ฏ ๐‘Ž1๐‘›
โ‹ฏ ๐‘Ž2๐‘›
]
โ‹ฎ
๐‘Ž๐‘š๐‘›

show that the product ๐‘ฆ๐ด 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)
2๐‘ฅ1 โˆ’ 3๐‘ฅ2 + 5๐‘ฅ3 = 7
9๐‘ฅ1 โˆ’ ๐‘ฅ2 + ๐‘ฅ3 = โˆ’1
๐‘ฅ1 + 5๐‘ฅ2 + 4๐‘ฅ3 = 0

1

Matrices sheet
b)
4๐‘ฅ1
โˆ’ 3๐‘ฅ3 + ๐‘ฅ4 = 1
5๐‘ฅ1 + ๐‘ฅ2
โˆ’ 8๐‘ฅ4 = 3
2๐‘ฅ1 โˆ’ 5๐‘ฅ2 + 9๐‘ฅ3 โˆ’ ๐‘ฅ4 = 0
3๐‘ฅ2 โˆ’ ๐‘ฅ3 + 7๐‘ฅ4 = 2
4)

Let ๐ด = [

3 1
] ,
5 2

๐ต=[

2 โˆ’3
],
4 4

๐ถ=[

6
4
]
โˆ’2 โˆ’1

verify that
๐‘Ž) (๐ดโˆ’1 )โˆ’1 = ๐ด

๐‘) (๐ต๐‘‡ )โˆ’1 = (๐ตโˆ’1 )๐‘‡

๐‘) (๐ด๐ต)โˆ’1 = ๐ตโˆ’1 ๐ดโˆ’1
5)

๐‘‘) (๐ด๐ต๐ถ)โˆ’1 = ๐ถ โˆ’1 ๐ตโˆ’1 ๐ดโˆ’1

Use Matrix row operations to solve the following systems, also find๐ดโˆ’1 of their
coefficient matrices .
a)
๐‘ฅ1 + 2๐‘ฅ2 + 3๐‘ฅ3 = 5
2 ๐‘ฅ1 + 5๐‘ฅ2 + 3๐‘ฅ3 = 3
๐‘ฅ1
+8๐‘ฅ3 = 17
b)
5 ๐‘ฅ1 + 3๐‘ฅ2 + 2๐‘ฅ3 = 4
3 ๐‘ฅ1 + 3๐‘ฅ2 + 2๐‘ฅ3 = 2
๐‘ฅ2 + ๐‘ฅ3 = 5

6)

Find all values of a, b, and c for which A is symmetric
2 ๐‘Ž โˆ’ 2๐‘ + 2๐‘
๐ด = [3
5
0
โˆ’2

7)

2๐‘Ž + ๐‘ + ๐‘
๐‘Ž+๐‘ ]
7

Find all values of a and b for which A and B are both not invertible.
๐ด=[

๐‘Ž+๐‘โˆ’1 0
]
0
3

5
๐ต=[
0

0
]
2๐‘Ž โˆ’ 3๐‘ โˆ’ 7

2


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