Consistency requires the two ranks to agree. Uniqueness additionally requires rank n, leaving no free variables. Thus r₁=r₂=n, B. Equal ranks below n give infinitely many solutions, whereas unequal ranks make the system inconsistent.
More Linear Algebra questions
GATE 2020 AG Q24– One eigenvalue of pmatrix3&2\\2&1pmatrix is 4.236. Find the other (three decimal places).
GATE 2026 AG Q14– Square matrices A,B of the same order satisfy (AB)=-80 and A=5. Find (BT).
GATE 2026 AG Q37– A 2×2 matrix has determinant 1.6 and eigenvalue sum −2.8. Find its eigenvalues.
GATE 2025 AG Q11– If a, b, c are the roots of 2x3-3x2+px-1=0 and two roots sum to 1,find p.
GATE 2025 AG Q12– If pmatrixx&-3\\2&x-5pmatrix is singular, find the sum of possible x values.
GATE 2024 AG Q36– Given P=pmatrix2&3&7\\4&1&5pmatrix and Q=pmatrix8&11&9\\12&6&13pmatrix,find PTQ.