Subject : Linear Algebra Assignment #2
Course code: MTH231 Total Marks: 20
Question#1
a) Find the inverse of the matrix ๐ด= 1 0 0
1 1 0
1 1 1 and ๐ต= 1 0 โ2
โ3 1 4
2โ3 4 , if it exists.
b) Find LU factorization of the matrix ๐ด= 1โ1 2
1โ3 1
3 7 5
c) Let
๐ฃ1= โ2
0
6 , ๐ฃ2= โ2
3
3 , ๐ฃ3= 0
โ5
5 and ๐= โ6
1
17
Determine if p is in the columns space of ๐ด ๐ฃ1 ๐ฃ2 ๐ฃ3 ? Is p is in the Null space of A?
Question#2
a) Find bases for Col A and Nul A, and then state the dimensions of these subspaces for the matrix
๐ด= 1โ2โ1 5 4
2โ1 1 5 6
โ20โ2 1 โ6
3 1 4 1 5
b) Use a determine to decide if ๐๐,๐๐,๐๐ are linearly independent, when
๐ฃ1= 4
6
โ7 , ๐ฃ2= โ7
0
2 , ๐ฃ3= โ3
โ5
6
c) Let ๐:๐
3โ๐
3 be the linear transformation determined by the matrix ๐0 0
0๐0
0 0 ๐ where a,b
and c are positive numbers. Let S be the unit ball, whose bounding surface has the equation
๐ฅ1
2+๐ฅ2
2+๐ฅ3
2= 1. Show that ๐ ๐ is bounded by the ellipsoid with the equation ๐ฅ1
2
๐2+๐ฅ2
2
๐2+๐ฅ3
2
๐2=
1.
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