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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 = [โ1 2] ,
1 1
6 1
๐ธ = [โ1 1
4 1

4 โ1
๐ต=[
],
0 2

1
๐ถ=[
3

4 2
],
1 5

1
๐ท = [โ1
3

5 2
0 1]
2 4

3
2]
3

Compute the following (If possible).

2)

โ3(๐ท + 2๐ธ)

(4๐ธ โ 2๐ท)

(2๐ด๐ + ๐ถ)

(๐ท๐ + ๐ธ ๐ )

1
1
( ๐ถ ๐ โ ๐ด)
2
4

(๐ต โ ๐ต๐ )

(2๐ธ ๐ โ 3๐ท๐ )

(๐ด๐ต)

(๐ด๐ต)C

๐ด(๐ต๐ถ)

(๐ถ ๐ ๐ต)๐ด๐

(๐ต๐ด๐ โ 2๐ถ)๐

๐ต๐ (๐ถ๐ถ ๐ โ ๐ด๐ ๐ด)๐

๐ท๐ ๐ธ ๐ โ (๐ธ๐ท)๐

(โ๐ด๐ถ)๐ + 5๐ท๐

let ๐ฆ = [๐ฆ1

๐11
๐ฆ2 โฆ โฆ โฆ ๐ฆ๐ ] , ๐ด = [ ๐21
โฎ
๐๐1

๐12
๐22
โฎ
๐๐2

โฏ ๐1๐
โฏ ๐2๐
]
โฎ
๐๐๐

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 โ 3๐ฅ2 + 5๐ฅ3 = 7
9๐ฅ1 โ ๐ฅ2 + ๐ฅ3 = โ1
๐ฅ1 + 5๐ฅ2 + 4๐ฅ3 = 0

1

Matrices sheet
b)
4๐ฅ1
โ 3๐ฅ3 + ๐ฅ4 = 1
5๐ฅ1 + ๐ฅ2
โ 8๐ฅ4 = 3
2๐ฅ1 โ 5๐ฅ2 + 9๐ฅ3 โ ๐ฅ4 = 0
3๐ฅ2 โ ๐ฅ3 + 7๐ฅ4 = 2
4)

Let ๐ด = [

3 1
] ,
5 2

๐ต=[

2 โ3
],
4 4

๐ถ=[

6
4
]
โ2 โ1

verify that
๐) (๐ดโ1 )โ1 = ๐ด

๐) (๐ต๐ )โ1 = (๐ตโ1 )๐

๐) (๐ด๐ต)โ1 = ๐ตโ1 ๐ดโ1
5)

๐) (๐ด๐ต๐ถ)โ1 = ๐ถ โ1 ๐ตโ1 ๐ดโ1

Use Matrix row operations to solve the following systems, also find๐ดโ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 โ 2๐ + 2๐
๐ด = [3
5
0
โ2

7)

2๐ + ๐ + ๐
๐+๐ ]
7

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

๐+๐โ1 0
]
0
3

5
๐ต=[
0

0
]
2๐ โ 3๐ โ 7

2