Tutorial MTH 3201Linear Algebras

Tutorial 0

1. Determine whether the following matrices are in Reduced Row Echelon form, Row Echelon Form, or not in both forms.

a) b)

c) d)

1 0 0

0 0 0

0 0 1

0 1 0

1 0 0

0 0 0

1 0 0

0 0 1

0 0 0

1 1 0

0 1 0

0 0 0

Reduced Row Echelon formBoth Not

Both Not

Row Echelon form

1. Determine whether the following matrices are in Reduced Row Echelon form, Row Echelon Form, or not in both forms.

e) f)

g) h)

1 0 0

0 1 0

0 2 0

1 5 3

0 1 1

0 0 0

1 3 4

0 0 1

0 0 0

1 3 0 2 0

1 0 2 2 0

0 0 0 0 1

0 0 0 0 0

Row Echelon form

Both Not

Row Echelon form

Row Echelon form

Reduced Row Echelon form• Every leading coeff. 1 ( not means identity )• If zero – lower part of matrix• Other elements in a column 1 must be zero• Leading 1 in the upper row is located at the left of the 1 in

the lower row• Eg: 1 0 1/ 2 0

0 1 1/ 3 0

0 0 0 1

Row Echelon form• All non zero row at least one non zero• Leading at the left.• Eg:

1 2

3

1

0 1

0 0 1

a a

a

BACK

2 1R R

2. Solve the following System of Linear Equations by using Gaussian Elimination Method.

3 13R R [1 1 20 −1 53 −7 4|

89

10 ][ 1 1 2−1 −2 33 −7 4| 8

110 ]

[1 1 20 1 −50 −10 −2|

8−9−14 ]

𝑅2 𝑋 (−1)

[1 1 20 1 −50 0 −52|

8−9−104 ]→

𝑅3+10𝑅2

y 5z=9, y=1 x + y + 2z = 8, x =3

, y=1, x =3[1 1 20 1 −50 0 1 | 8

−92 ]→

𝑅3 𝑋−152

3. Solve the system of linear equations in Question (2) by

using Gauss-Jordan Elimination Method.

[1 1 20 1 −50 0 1 | 8

−92 ]→𝑅2+5𝑅3[1 1 2

0 1 00 0 1|

812 ]→𝑅1−𝑅2[1 0 2

0 1 00 0 1|

712 ]

→𝑅1−2𝑅3[1 0 00 1 00 0 1|

312 ] , y=1, x =3

2. Solve the following System of Linear Equations by using Gaussian Elimination Method.

1 1 2 1 1

0 1 2 0 0

0 0 0 0 0

0 0 0 0 0

, 2 , , .

Ans

x s y t z t w s

ERO

4. By using Elementary Row Operations on the Augmented Matrix [A|I], find the inverse of the following matrix A.

[1/5 1/5 −2 /51/5 1/5 1/101/5 −4 /5 1/10 |1 0 0

0 1 00 0 1 ] [1 1 −2

0 1 01 −4 1/2|

5 0 00 1 −10 0 5 ]→

𝑅1 𝑋 5

→𝑅3−𝑅1 →

𝑅3+5𝑅2

→ / (5/2)

→𝑅1+2𝑅3

𝑅2−𝑅3

𝑅3 𝑋 5

[1 1 −20 1 00 −5 5/2|

5 0 00 1 −1−5 0 5 ] [1 1 −2

0 1 00 0 5/2|

5 0 00 1 −1−5 5 0 ]

[1 1 −20 1 00 0 1 | 5 0 0

0 1 −1−2 2 0 ] [1 1 0

0 1 00 0 1|

1 4 00 1 −1−2 2 0 ]

→𝑅1−𝑅2 [1 0 0

0 1 00 0 1|

1 3 10 1 −1−2 2 0 ]

(𝑎)

4. By using Elementary Row Operations on the Augmented Matrix [A|I], find the inverse of the following matrix A.

1

1 0 0 0

1 10 0

3 31 1

0 05 5

1 10 0

7 7

A

1

3 1 11

4 4 43

0 1 021 0 0 0

1 1 11

4 4 4

A

−4 /5 3 /5 −1/53/2 0 11/2 0 04 /5 2/5 1/5

1/5000

−1 /5

Note: Row Equivalent= Each matrix is row equivalent to a unique reduce achelon form matrix

. .

1 2 3 1 0 0

1 4 1 0 1 0

2 1 9 0 0 1

E R OA

Reduce Row echelon form

. .

1 0 5 1 0 0

0 2 2 0 1 0

1 1 4 0 0 1

E R OB

Reduce Row echelon form

A=B : same reduce achelon form matrix

5. …. Find a sequence of elementary row operatons that generates B from A.

. .

1 2 3 1 0 0

1 4 1 0 1 0

2 1 9 0 0 1

E R OA

2 3

1 0 0

0 1 1

0 0 1

R R

1 35

1 0 5

0 1 1

0 0 1

R R

3 1

3 2

1 0 5

0 1 1

1 1 4

R RR R

22

1 0 5

0 2 2

1 1 4

R B

Exist solution (consistent) with condition bi (1 ≤ i ≤ 3) …1…,…2…, and …3… respectively.

6. Find the condition of bi (1 ≤ i ≤ 3) such that the following systems are Consistent (in which the solution exists).

1

. .2

3

1 2 5 1 0 0 ...1...

4 5 8 0 1 0 ...2...

0 0 1 ...3...3 3 3

E R O

b

b

b

Exist solution (consistent) with condition bi (1 ≤ i ≤ 3) …1…,…2…, and …3… respectively.

6. Find the condition of bi (1 ≤ i ≤ 3) such that the following systems are Consistent (in which the solution exists).

1

. .2

3

1 2 1 1 0 0 ...1...

4 5 2 0 1 0 ...2...

0 0 1 ...3...4 7 4

E R O

b

b

b