Elementary Transformations and Rank
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Direct answer
Rank is the one number about a matrix that elementary operations cannot change. Interchanging two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another (and the same three moves on columns) leave the rank untouched, so the working method is: reduce the matrix to row echelon form by elementary row operations and count the nonzero rows — that count is the rank. Equivalently, rank is the order of the largest square submatrix with a nonzero determinant, and for an m × n matrix it never exceeds min(m, n). Products can only lose rank: rank(AB) ≤ min(rank A, rank B), and Sylvester's inequality refines the loss: rank(AB) ≥ rank A + rank B - n when A has n columns. A square matrix has full rank exactly when its determinant is nonzero.
What you must remember
- Elementary operations: row swap, nonzero scaling of a row, adding a multiple of a row to another (plus the three column analogues) — none changes rank.
- Echelon method: reduce to row echelon form (zeros below leading entries, each leading entry right of the one above); rank = number of nonzero rows.
- Equivalent characterisations: rank = order of the largest nonsingular minor; rank = number of linearly independent rows = number of linearly independent columns (row rank equals column rank).
- Basic bounds: rank(A) ≤ min(m, n); rank(A + B) ≤ rank A + rank B; rank(A^T) = rank(A).
- Product inequality: rank(AB) ≤ min(rank A, rank B); Sylvester: rank(AB) ≥ rank A + rank B - n, with n the shared inner dimension.
- Full rank characterisation: for an n × n matrix, rank n ⟺ det ≠ 0 ⟺ invertible; rank deficiency by one means every (n × n) determinant vanishes but some (n - 1) minor survives.
- Row operations cannot change row dependence: if rows are linearly dependent before, they remain dependent after — the theorem behind counting nonzero rows as the rank.
Reducing one matrix to echelon form
Find the rank of A with rows (1, 2, 1), (2, 3, 3), (1, 1, 2). Apply R2 → R2 - 2R1: the second row becomes (0, -1, 1). Apply R3 → R3 - R1: the third row becomes (0, -1, 1). Now R3 → R3 - R2 reduces the third row to (0, 0, 0). The echelon form has two nonzero rows, so rank(A) = 2. Cross-check with minors: every 3 × 3 determinant is zero (the third row was a combination of the first two), while the 2 × 2 minor from columns 1 and 2 of rows 1 and 2 equals 1 × 3 - 2 × 2 = -1 ≠ 0, certifying rank 2 again. Note the practical grammar of the reduction: eliminate below the first pivot, move to the next column, repeat — and a column of zeros under a pivot means the pivot shifts right, not down. The same procedure applied to the augmented matrix, not the coefficient matrix alone, is precisely how consistency of linear systems is decided, which is why this page is the machinery behind every system question.
How the exam frames it
JEE Main asks for the rank of small matrices (2 × 3, 3 × 3 with an obvious dependency) as numerical-value questions, and for the value of k making a given rank drop — find k so that a 3 × 3 matrix has rank 2 amounts to setting its determinant to zero. Advanced tests the inequalities: given rank A and rank B, bound rank(AB) (Sylvester's inequality supplies the sharp lower end), or ask whether rank(A + B) can exceed the sum of ranks (never). The characteristic slips: counting the rows of the original matrix instead of the echelon form; stopping halfway — a matrix whose second row is (0, 0, 0) but whose third row is nonzero needs a row swap before counting; dividing by a pivot that happens to be zero (scan the column first, swap if needed); and confusing elementary operations with arbitrary row operations — multiplying a row by zero is not elementary and destroys rank information. Matrices and determinants form a named unit in both syllabi, and rank is the concept the exam uses to link determinants with systems.
Frequently asked questions
How is the rank of a matrix found by elementary transformations?
Reduce to row echelon form using row swaps, nonzero scalings and row additions; the number of nonzero rows in the echelon form is the rank.
Which operations leave rank unchanged?
All six elementary operations — three on rows and three on columns — preserve rank; multiplying a row by zero is not elementary and can change it.
What is the largest possible rank of an m × n matrix?
min(m, n); rank equals the maximum number of linearly independent rows or columns, and there are only that many of each to choose from.
How does rank behave under matrix multiplication?
rank(AB) ≤ min(rank A, rank B), and Sylvester's inequality gives the lower bound rank(AB) ≥ rank A + rank B - n where n is the shared inner dimension.
When does a square matrix have full rank?
Exactly when its determinant is nonzero — equivalently when it is invertible; a zero determinant with a surviving (n - 1) minor certifies rank n - 1.