a. Ax=b is inconsistent (i.e., no solution exists) if and only if rank[A]<rank[A∣b].
b. Ax=b has an unique solution if and only if rank[A]=rank[A∣b]=n.
c. Ax=b has infinitely many solutions if and only if rank[A]=rank[A∣b]<n.
i.
A=⎣⎡012−64−501−34⎦⎤ Swap rows 1and 2
⎣⎡120−6−540−314⎦⎤R3=R3+(1/2)R1
⎣⎡1200−54−5/2−315/2⎦⎤ R3=R3+(5/8)R2
The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.
Since rank[A]=rank[A∣b]=3=n, then Ax=b has an unique solution.
ii.
A=⎣⎡528−3391−1−3⎦⎤ R1=R1/5
⎣⎡128−3/5391/5−1−3⎦⎤R2=R2−2R1
⎣⎡108−3/521/591/5−7/5−3⎦⎤ R3=R3−8R1
⎣⎡100−3/521/569/51/5−7/5−23/5⎦⎤ R2=(5/21)R2
⎣⎡100−3/5169/51/5−1/3−23/5⎦⎤ R1=R1+(3/5)R2
⎣⎡1000169/50−1/3−23/5⎦⎤ R3=R3−(69/5)R2
⎣⎡1000100−1/30⎦⎤ The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 2.
⎣⎡528−3391−1−3702⎦⎤ R1=R1/5
⎣⎡128−3/5391/5−1−37/502⎦⎤R2=R2−2R1
⎣⎡108−3/521/591/5−7/5−37/5−14/52⎦⎤ R3=R3−8R1
⎣⎡100−3/521/569/51/5−7/5−23/57/5−14/5−46/5⎦⎤ R2=(5/21)R2
⎣⎡100−3/5169/51/5−1/3−23/57/5−2/3−46/5⎦⎤ R1=R1+(3/5)R2
⎣⎡1000169/50−1/3−23/51−2/3−46/5⎦⎤ R3=R3−(69/5)R2
⎣⎡1000100−1/301−2/30⎦⎤ The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 2.
Since rank[A]=rank[A∣b]=2<3=n, then Ax=b has infinitely many solutions.
iii.
A=⎣⎡−8012062240⎦⎤ R1=−R1/8
⎣⎡1012062−1/440⎦⎤R3=R3−12R1
⎣⎡100062−1/443⎦⎤ R2=R2/6
⎣⎡100012−1/42/33⎦⎤ R3=R3−2R2
⎣⎡100010−1/42/35/3⎦⎤ R3=(3/5)R3
⎣⎡100010−1/42/31⎦⎤ R1=R1+(1/4)R3
⎣⎡10001002/31⎦⎤ R2=R2−(2/3)R3
⎣⎡100010001⎦⎤
The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.
Since rank[A]=rank[A∣b]=3=n, then Ax=b has an unique solution.
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