LINEAR ALGEBRA QUESTION A is a matrix which consists of 5 rows and 4 columns row
ID: 3122623 • Letter: L
Question
LINEAR ALGEBRA QUESTION
A is a matrix which consists of 5 rows and 4 columns
row1 : 2 - 1 3 4
row 2: 2 c 0 6
row 3: -2 1 (c-5) -4
row 4: 2 c 0 (c+5)
row 5: 0 (c+1) (c-5) 2
B is a matrix which consists of only one column and 5 rows.
column 1: 1
1
-1
2
0
x is a matrix which consists of 4 rows and one column
column1: x
y
z
w
determine all values of c for which the system AX = b has (i) no solution, (ii) infinitely many solutions, (iii) unique solution. Write the solutions in cases (ii) and (iii).
Explanation / Answer
AX= B
2 -1 3 4 / 1
2 c 0 6 / 1
-2 1 (c-5) -4 / -1
2 c 0 (c+5) / 2
0 (c+1) ( c-5) 2 / 0
row 4= row 3 and row 4 and row3= row 2+row 3 and Row2= row2-row 1 , row1= row1/2
1 -1/2 3 /2 2 / 1/2
0 c+1 -3 2 / 0
0 1+c (c-5) 2 / 0
0 c+2 c-5 (c+1) / 1
0 (c+1) ( c-5) 2 / 0
Now Row5= row4-row 5 and row4= row4-row3
1 -1/2 3 /2 2 / 1/2
0 c+1 -3 2 / 0
0 1+c (c-5) 2 / 0
0 1 0 (c-1) / 1
0 -1 0 c-1 / 1
row 5= row5-row4 and row3-row3-row2
1 -1/2 3 /2 2 / 1/2
0 c+1 -3 2 / 0
0 0 (c-2) 0 / 0
0 1 0 (c-1) / 1
0 -2 0 0 / 0
row4=-2Row 4+row2
1 -1/2 3 /2 2 / 1/2
0 c+1 -3 2 / 0
0 0 (c-2) 0 / 0
0 0 0 -2 (c-1) / 1
0 -2 0 0 / 0
-2(c-1) w=1
-2(c-1)=1
c-1=-1/2
(iii) c=1/2 is the unique solution i.e y=0,z=0,w=1,x=-3/2
(i) when -2(c-1)=0 the system will have no solution
thus at c=1 there will be no solution
(ii)at c-2=0 i.e c=2 the system will have infinite solution
z=0, w=-1/2,y=0,x=3/2
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