Determinants, row operations and invertibility

Objectives, prerequisites and scope

Compute determinants efficiently, track row-operation effects and use a zero determinant correctly. Prerequisite: matrix multiplication and square matrices. This lesson covers the official Determinants heading and supplies tools for systems and eigenvalues. Unless stated otherwise, matrices are real and square when their determinant is taken.

1. The determinant is a scalar, not a matrix

For A=[[a,b],[c,d]], det(A)=ad−bc. A determinant is defined for square matrices. For [[2,3],[1,4]] it is 8−3=5. A rectangular matrix has rank but no ordinary determinant; do not label its determinant zero.

For a 3×3 matrix, expand along a row or column. The minor Mᵢⱼ is the determinant after deleting row i and column j; the cofactor Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ. The sign pattern begins +,−,+ in the first row and alternates. Expansion along row i gives det(A)=ΣⱼaᵢⱼCᵢⱼ. Choose a row or column with zeros to reduce work.

Worked example: A=[[1,2,0],[3,4,0],[5,6,7]]. Expanding the last column gives det(A)=7 det([[1,2],[3,4]])=7(4−6)=−14. The third-column first two entries are zero, and the surviving cofactor sign is positive. Expanding another row must give the same scalar.

2. Row operations have different effects

Swapping two rows changes the sign. Multiplying one row by c multiplies the determinant by c. Replacing Rᵢ by Rᵢ+cRⱼ for i≠j leaves it unchanged. The corresponding column rules also hold. A common error is treating all elimination steps as determinant-preserving: swaps and scalings are not.

A triangular matrix has determinant equal to the product of its diagonal entries. Thus elimination to triangular form is often faster than cofactor expansion. Record the accumulated sign and scaling separately.

Worked example: A=[[0,2,1],[1,3,1],[2,8,3]]. Swap R₁ and R₂ once, then use R₃←R₃−2R₁ to obtain [[1,3,1],[0,2,1],[0,2,1]]. Finally R₃←R₃−R₂ yields a zero row. Its determinant is zero, hence so is det(A); the swap sign cannot change zero. This example also exposes dependent rows.

Nonzero example: A=[[2,1,0],[4,3,1],[0,2,5]]. R₂←R₂−2R₁ gives [[2,1,0],[0,1,1],[0,2,5]]. R₃←R₃−2R₂ gives diagonal entries 2,1,3 in an upper triangular matrix. No swap or scaling occurred, so det(A)=6.

3. Structural identities and their assumptions

For n×n matrices, det(AB)=det(A)det(B), det(Aᵀ)=det(A) and det(I)=1. Therefore det(cA)=cⁿdet(A): every one of n rows was scaled. This is not c det(A) unless n=1 or special values happen to agree. If A is invertible, det(A⁻¹)=1/det(A).

There is no general additive identity det(A+B)=det(A)+det(B). For 2×2 I, det(I+I)=det(2I)=4 but det(I)+det(I)=2. The determinant is linear in one row while other rows are held fixed, not linear in the whole matrix.

Equal rows, proportional rows, a zero row or any linear dependence among rows force determinant zero. Dependence among columns does likewise. Conversely, for a square matrix, determinant zero means the rows and columns are dependent. A matrix can have a zero diagonal entry but a nonzero determinant: [[0,1],[1,0]] has determinant −1.

4. Invertibility and the 2×2 inverse

A square matrix is invertible exactly when det(A)≠0. One direction follows from det(AA⁻¹)=det(I)=1. For 2×2 A with ad−bc≠0,

A⁻¹=(1/(ad−bc))[[d,−b],[−c,a]].

For [[2,3],[1,4]], the inverse is (1/5)[[4,−3],[−1,2]]. Multiplying verifies I. If ad−bc=0, do not divide by zero or invent an inverse. Larger matrices can be inverted by reducing [A|I] to [I|A⁻¹] when possible; explicit inverse computation is usually unnecessary just to solve one system.

Parameter example: A(t)=[[1,2],[3,t]] has determinant t−6. It is invertible for every t≠6 and singular at t=6. Determinant zero alone does not tell whether Ax=b has no solution or infinitely many; b must also be checked. The next lesson makes that distinction using augmented rank.

5. Solving checklist and pitfalls

First confirm squareness. Look for triangular structure, zeros or row dependence. Choose expansion or elimination; preserve a record of row swaps and row scalings. Check whether a claimed identity concerns products or sums. When parameters occur in a pivot or denominator, split off their zero cases before division. Keep det(A), rank(A) and trace(A) distinct: determinant is a scalar product-like invariant, rank counts independent directions, and trace sums diagonal entries. Their connections to eigenvalues are developed separately.

Original practice: six single-correct MCQs

These are original questions, not previous-year questions. Choose exactly one option.

LD01. What is det([[2,3],[1,4]])?

A. 11 B. 5 C. 8 D. −5

Answer: B

A: The cross product must be subtracted, not added.

B: 2×4−3×1=5.

C: This is only the main-diagonal product.

D: This reverses the subtraction.

LD02. A is 3×3 with det(A)=2. What is det(2A)?

A. 4 B. 8 C. 16 D. 6

Answer: C

A: Scaling the entire matrix scales all three rows.

B: The multiplier is 2³, but it must also multiply det(A).

C: det(2A)=2³det(A)=16.

D: Determinant scaling uses a power, not matrix size times det(A).

LD03. B is obtained from A by replacing row 2 with row 2+5 row 1. If det(A)=−3, find det(B).

A. −15 B. 3 C. 2 D. −3

Answer: D

A: Adding a multiple of another row does not scale the determinant.

B: A row swap would reverse the sign; the stated row replacement does not.

C: The coefficient 5 is not added to the determinant.

D: A row replacement of this form leaves the determinant unchanged.

LD04. For what real k is [[1,2],[3,k]] singular?

A. 0 B. 3 C. 6 D. 2

Answer: C

A: The determinant is k−6, not k.

B: At k=3 the determinant is −3, which is nonzero.

C: Singularity requires k−6=0.

D: At k=2 the determinant is −4.

LD05. A and B are 2×2 with det(A)=−2 and det(B)=3. What is det(A⁻¹B)?

A. −6 B. −3/2 C. −2/3 D. 3/2

Answer: B

A: The first factor is A inverse, whose determinant is −1/2.

B: det(A⁻¹B)=det(B)/det(A)=−3/2.

C: This reverses the ratio.

D: The determinant of A is negative, so the result is negative.

LD06. A=[[1,2,0],[0,3,4],[0,0,−2]]. What is det(A)?

A. 2 B. −6 C. 24 D. 0

Answer: B

A: This is the diagonal sum, not the determinant.

B: For a triangular matrix multiply diagonal entries: 1×3×(−2)=−6.

C: Off-diagonal entries do not enter the triangular determinant product.

D: Zero entries below the diagonal do not force a zero determinant.

Source and scope

Aligned to Determinants and matrix invertibility in the official 2026 syllabus. System consistency and eigenvalue applications follow separately. Original practice; no exhaustive readiness claim.

Official GATE 2026 CS syllabus (verified 2026-10-14).