Matrices and Determinants

Key determinant facts for GATE: det(AB) = det(A)det(B); det(A^T) = det(A); det(kA) = k^n·det(A) for an n×n matrix; det(A^-1) = 1/det(A). A row/column swap flips the sign; adding a multiple of one row to another leaves det unchanged. det(A) = product of eigenvalues. A triangular matrix's determinant is the product of its diagonal entries. Memory aid: 'SWAP-FLIP, SCALE-power-n, INVERSE-reciprocal.' For a 2×2 [[a,b],[c,d]], det = ad − bc. A matrix is singular (non-invertible) iff det = 0, which means rank < n and at least one eigenvalue is 0. These properties let you avoid full cofactor expansion under exam time pressure.

Rank is the single most expressive number you can attach to a matrix — it tells you, in one integer, how much "real information" the matrix carries. In GATE Engineering Mathematics, rank ties together row operations, solvability of linear systems, invertibility, determinants, and the rank-nullity theorem. Master rank and you have understood half of linear algebra.

Definition: The rank of a matrix A, written rank(A) or ρ(A), is the maximum number of linearly independent rows of A; equivalently, the maximum number of linearly independent columns of A; equivalently, the number of non-zero rows in any row echelon form of A; equivalently, the order of the largest non-zero minor of A.

The fact that all four definitions agree is itself a deep theorem — the row rank equals the column rank, even for a non-square matrix. You can choose whichever definition is most convenient for the problem at hand.

How big can rank get?

For an m × n matrix, rank cannot exceed the number of rows or the number of columns, so

rank(A) ≤ min(m, n).

When rank(A) equals min(m, n) we say A has full rank:

• If m ≤ n and rank = m, A has full row rank — every row is independent.
• If n ≤ m and rank = n, A has full column rank — every column is independent.
• For a square matrix (m = n), full rank means rank = n; this is exactly the condition for invertibility.

For a non-zero matrix, rank is at least 1. The zero matrix is the only matrix with rank 0. A rank-1 matrix has a very special structure: it can always be written as an outer product uvᵀ of two non-zero column vectors. Every row is a scalar multiple of the same vector.

Why it matters — solvability of Ax = b

The most important application of rank is the Rouché-Capelli criterion for the system Ax = b where A is m × n and [A | b] is the augmented matrix:

• The system is inconsistent (no solution) iff rank(A) < rank([A | b]).
• The system is consistent (has at least one solution) iff rank(A) = rank([A | b]).
• The consistent system has a unique solution iff rank(A) = rank([A | b]) = n.
• The consistent system has infinitely many solutions iff rank(A) = rank([A | b]) < n. The solution space has dimension n − rank(A) (this many free parameters).

For a square n × n matrix, rank(A) = n is equivalent to det(A) ≠ 0, which is equivalent to A being invertible, which is equivalent to Ax = 0 having only the trivial solution x = 0.

How to compute rank — Gaussian elimination

The fastest, most reliable method on the GATE exam is:

Step 1: Apply elementary row operations to reduce A to row echelon form. The three legal operations are: swap two rows; multiply a row by a non-zero scalar; add a scalar multiple of one row to another. Each of these preserves rank.
Step 2: Count the number of non-zero rows. That count is rank(A). Equivalently, count the pivots (the first non-zero entry of each non-zero row).

This is fast because elementary row operations are mechanical and the answer falls out instantly once the matrix is in echelon form. You do not need reduced echelon form — plain echelon form is enough.

The other method — finding the largest non-zero minor — is useful for small matrices and for theoretical proofs but is slow for anything larger than 3 × 3.

The Rank-Nullity Theorem

For any m × n matrix A:

rank(A) + nullity(A) = n

where the nullity is the dimension of the null space {x : Ax = 0}, that is, the number of free parameters in the general solution of Ax = 0. This identity is your most powerful shortcut: if you know one of rank or nullity, you know the other.

For example, if A is 4 × 7 and rank(A) = 3, then nullity(A) = 7 − 3 = 4. The homogeneous system Ax = 0 has a 4-parameter family of solutions.

Rank inequalities you should memorise

• rank(AB) ≤ min(rank(A), rank(B)) — multiplication cannot increase rank.
• rank(A + B) ≤ rank(A) + rank(B) — subadditive under addition.
• rank(A) = rank(Aᵀ) = rank(AᵀA) = rank(AAᵀ) — transpose and Gram matrices have the same rank.
• For an outer product, rank(uvᵀ) = 1 whenever u, v ≠ 0.
• rank(A) is invariant under multiplication by an invertible matrix on either side.

The first inequality is the source of many GATE traps. If A is 3 × 5 with rank 2 and B is 5 × 4 with rank 3, then rank(AB) ≤ 2 — multiplication is capped by the smaller rank.

Worked example

Question: Find the rank of A = [[1, 2, 3], [2, 4, 7], [3, 6, 10]].

Solution:
Step 1: Row-reduce. R₂ → R₂ − 2 R₁ gives [0, 0, 1]. R₃ → R₃ − 3 R₁ gives [0, 0, 1].
Step 2: R₃ → R₃ − R₂ gives [0, 0, 0]. The matrix is now [[1, 2, 3], [0, 0, 1], [0, 0, 0]].
Step 3: Count non-zero rows. Two non-zero rows.
Conclusion: rank(A) = 2. Since A is 3 × 3 and rank ≠ 3, det(A) = 0 and A is singular. The homogeneous system Ax = 0 has nullity 3 − 2 = 1, that is, one free parameter.

Why it matters in practice

Rank is the dimension of the column space (image) of A as a linear map. When you build a least-squares regression in machine learning, the design matrix must have full column rank for the normal equations to give a unique solution. When you compress a high-resolution image with SVD, you are throwing away small singular values — equivalently, approximating the image by a lower-rank matrix. The same matrix-rank ideas reappear in network theory, control systems, and computer graphics.

Real-world example: PageRank, the original algorithm behind Google, treats the web as a giant sparse matrix and computes a dominant eigenvector. The numerical stability of that calculation depends fundamentally on whether the matrix has full rank — and many years of mathematical research go into "regularising" near-rank-deficient matrices in industry.

Common misconception: Students confuse rank with the number of non-zero rows of the original matrix. Wrong — the rows of the original matrix may be linearly dependent even when none of them is zero. You must reduce to echelon form first. Three identical non-zero rows give rank 1, not rank 3.

Another misconception: rank(A + B) = rank(A) + rank(B). False in general — consider A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]]. Each has rank 1; their sum is the identity, which has rank 2. But A + (−A) has rank 0, not 2 × rank(A). Only the inequality rank(A + B) ≤ rank(A) + rank(B) holds.

Find det(A) where A is 3×3 with trace = 6 and you are told eigenvalues are in arithmetic progression with the smallest being 1. AP with smallest 1: eigenvalues λ, λ+d, λ+2d summing to trace = 6, so 3λ+3d = 6 → λ+d = 2 (the middle term). With smallest = 1, middle = 2, largest = 3. Then det(A) = product of eigenvalues = 1×2×3 = 6. This shows two shortcuts: trace = sum of eigenvalues, det = product of eigenvalues. Always use these to bypass cofactor expansion when eigenvalue information is given.

Cover the answer, recall, then check. 12 cards on the matrix/determinant facts GATE reuses constantly.

Q1. State the key multiplicative property of determinants.
A1. det(AB) = det(A)·det(B) (for square A, B of the same size). Hence det(Aᵏ) = det(A)ᵏ.

Q2. How does det(A) relate to det(Aᵀ) and det(kA)?
A2. det(Aᵀ) = det(A); det(kA) = kⁿ·det(A) for an n×n matrix (scalar multiplies every row).

Q3. What is det(A⁻¹)?
A3. 1/det(A), defined only when det(A) ≠ 0.

Q4. Give the formula for A⁻¹ using the adjugate.
A4. A⁻¹ = adj(A)/det(A), where adj(A) is the transpose of the cofactor matrix.

Q5. What is det(adj(A)) for an n×n matrix?
A5. det(adj(A)) = det(A)ⁿ⁻¹.

Q6. How does swapping two rows affect the determinant? Adding a multiple of one row to another?
A6. A row swap multiplies det by −1; adding a multiple of one row to another leaves det unchanged (row reduction to triangular form is the fast way to compute det).

Q7. When is a square matrix invertible (non-singular)?
A7. ⇔ det(A) ≠ 0 ⇔ full rank ⇔ rows/columns linearly independent ⇔ 0 is not an eigenvalue.

Q8. Define the rank of a matrix.
A8. The number of linearly independent rows (= independent columns) = number of nonzero rows in row-echelon form = size of the largest nonzero minor.

Q9. What is the determinant of a triangular matrix?
A9. The product of the diagonal entries.

Q10. Define symmetric, skew-symmetric, orthogonal, and idempotent matrices.
A10. Symmetric: Aᵀ=A. Skew-symmetric: Aᵀ=−A (diagonal all 0). Orthogonal: AᵀA=I (so A⁻¹=Aᵀ). Idempotent: A²=A.

Q11. What is the trace, and its cyclic property?
A11. Sum of diagonal entries. trace(AB) = trace(BA) even when AB ≠ BA; trace is invariant under similarity.

Q12. For n×n matrices, does (AB)⁻¹ = A⁻¹B⁻¹?
A12. No — (AB)⁻¹ = B⁻¹A⁻¹ (order reverses). Same reversal for transpose: (AB)ᵀ = BᵀAᵀ.

Matrices and determinants are the toolbox underneath every other Linear Algebra question in GATE CSE — eigenvalues, linear systems, and vector spaces all lean on determinant and rank facts. Direct questions (compute a determinant, a rank, an inverse, or apply a property) appear most years for 1–2 marks, and the same facts silently decide harder problems.

Why it matters

The determinant is a single number that tells you whether a matrix is invertible, how it scales volume, and whether a linear system has a unique solution. Rank tells you the true dimensionality of the map. Getting these instincts fast frees time for the reasoning-heavy questions.

Key results

Quantity	Result
det(AB)	det(A)·det(B)
det(kA), n×n	kⁿ·det(A)
det(A⁻¹)	1/det(A)
det(adj A), n×n	det(A)ⁿ⁻¹
A⁻¹	adj(A)/det(A)
(AB)⁻¹	B⁻¹A⁻¹
Triangular det	product of diagonal

Exam Tricks & Tips

• 🎯 Row-reduce to triangular to get a determinant fast — adding row multiples doesn't change it, and a swap only flips the sign.
• 🎯 det(kA) = kⁿ det(A) trips students who write k·det(A); always raise k to the matrix order n.
• 🎯 Order reverses for inverse and transpose of a product: (AB)⁻¹ = B⁻¹A⁻¹, (AB)ᵀ = BᵀAᵀ.
• 🎯 Rank = largest nonzero minor — a quick route when echelon reduction is messy.
• 🎯 Odd-order skew-symmetric ⇒ det = 0 (a free elimination in MCQs).
• ❌ Common mistake: assuming det(A + B) = det(A) + det(B). Determinant is not additive — only multiplicative over products.

Expected exam pattern

Compute a 3×3 determinant or rank; apply det(kA) or det(adj A); identify a matrix type (orthogonal/idempotent) from a property; or use invertibility to reason about a system. Often a stepping stone inside a larger question.

Quick recap

det(AB)=det A·det B, det(kA)=kⁿdet A, det(adj A)=det(A)ⁿ⁻¹, A⁻¹=adj(A)/det(A). Inverse/transpose of a product reverses order. Invertible ⇔ det ≠ 0 ⇔ full rank. Row operations: swap flips sign, add-multiple keeps det, so triangularize to compute.