The rank of a matrix is the number of non-zero rows in its row-echelon form,
equivalently the order of its largest non-vanishing minor, equivalently the maximum
number of linearly independent rows (or columns). Elementary row/column operations do
not change the rank, so a matrix is reduced by such operations and the surviving
non-zero rows are counted.
Consequences used in problems: a matrix has rank 1 exactly when every 2×2
minor vanishes — i.e. all rows are proportional; the given non-zero condition on
entries then forces relations among the unknowns. An n×n matrix has full rank
n iff its determinant is non-zero. When a matrix has repeated or proportional rows
(for example all rows equal), its rank drops accordingly (a 3×3 all-ones matrix
has rank 1). Reading the echelon form after the stated operations gives the rank
directly.
Matrix rank builds on the NCERT/CBSE Class 12 Mathematics "Matrices" and "Determinants" chapters and is an important topic for JEE Main, JEE Advanced and state CETs, extending the row-reduction techniques introduced there. "Rank of a matrix examples" and "matrices class 12 maths important questions" are common searches this concept addresses.
Reducing the matrix to echelon form by elementary row operations and counting the non-zero rows gives the rank directly.
ρ(A)=2.
R2→R2−2R1, R3→R3−3R1 gives rows [1,2,3],[0,−1,−2],[0,−1,−2]; then R3→R3−R2 gives a zero row. Two non-zero rows remain.
✓Final answer
ρ(A)=2
Row-reducing A
A=123235347
R2→R2−2R1: [2−2(1),3−2(2),4−2(3)]=[0,−1,−2].
R3→R3−3R1: [3−3(1),5−3(2),7−3(3)]=[0,−1,−2].
1002−1−13−2−2
R3→R3−R2: [0−0,−1−(−1),−2−(−2)]=[0,0,0].
1002−103−20
Two non-zero rows remain, so ρ(A)=2.
Check (independent recomputation via detA):detA=1(3×7−4×5)−2(2×7−4×3)+3(2×5−3×3)=1(21−20)−2(14−12)+3(10−9)=1−4+3=0. Since detA=0, ρ(A)<3. The top-left 2×2 minor 1223=3−4=−1=0, so a non-zero 2×2 minor exists — confirming ρ(A)=2 exactly.
✓Final answer
ρ(A)=2
Stopping after the first row operation and reading off 3 non-zero rows without checking whether a later row becomes a multiple of an earlier one — always fully reduce to echelon form before counting.