SIT292: Linear Algebra | Assignment

SIT292 LINEAR
ALGEBRA
ASSIGNMENT- 3
STUDENT ID
[Pick the date]

Question 1
Given matrix
(a) Row- rank of matrix
In order to determine the rank of the above matrix, it is essential to transform the matrix into
upper triangular matrix with the help of row operations.
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It can be seen from the above upper triangular matrix that there is three non-zero rows and hence,
the row- rank of matrix A is 3.
(b) Set of generators for the row space of A is highlighted below:
{ (1 2−3 0 ) ( 2 4−2 2 ) ( 36−4 3 ) }
(c) Basis for the row space of matrix A
It is apparent that all the three rows of the given matrix A are linearly independent because the
row rank of the matrix is three. Further, these three rows will make the basis for the respective
row spaces. Therefore, the basis of the row space is shown below:
{ (1 2−3 0 ) ( 2 4−2 2 ) ( 36−4 3 ) }
Therefore, the generator and the basis vectors are the same for the given problem.
Question 2
Given matrix
Let A=
[0 2 0
1 0 1
0 2 0 ]
(a) Eigenvalue
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Eigenvalues would be the roots of the matrix which would be determined by using the
determinant of ( A−λ I )=0
( A−λ I ) =
[ 0 2 0
1 0 1
0 2 0 ] −λ [ 1 0 0
0 1 0
0 0 1 ]
( A−λ I )=
[0 2 0
1 0 1
0 2 0 ]−
[ λ 0 0
0 λ 0
0 0 λ ]
( A−λ I )=
[−λ 2 0
1 −λ 1
0 2 −λ ]
det ( A−λ I )=
|−λ 2 0
1 −λ 1
0 2 −λ|
¿−λ ( λ2−2 )−2 (−λ−0 )+ 0(2+ λ)
¿−λ ( λ2−2 )+2 λ+0
¿−λ3+ 2 λ+ 2 λ
¿−λ3+ 4 λ
¿−λ ( λ2−4 )
¿−λ ( λ−2 ) ( λ+2 )
Now, put det ( A−λ I )=0
−λ ( λ−2 ) ( λ+2 ) =0
λ=0 , 2 ,−2
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Hence, the eigenvalues of the given matrix is 0 , 2 ,−2.
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(b) Eigenvector corresponding to eigenvalues
 Eigenvector corresponding to eigenvalues λ = 0
( A−λ I ) =
[ 0 2 0
1 0 1
0 2 0 ] −0 [ 1 0 0
0 1 0
0 0 1 ]
¿ [0 2 0
1 0 1
0 2 0 ]−
[0 0 0
0 0 0
0 0 0 ]
¿ [ 0 2 0
1 0 1
0 2 0 ]
Reduce the matrix into row echelon form
[ (a ⋯ b
0 ⋱ ⋮
0 0 c ) ]
¿ [ 1 0 1
0 2 0
0 0 0 ]
Reduce the matrix to reduced row echelon form
[ (1 ⋯ b
0 ⋱ ⋮
0 0 1) ]
¿ [ 1 0 1
0 1 0
0 0 0 ]
Now,
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