MathsMediumClass 12

Transpose — Properties

Matrices

JEE Qs

Hard

min

Master the property (AB)' = B'A' as it is the most common source of errors and forms the basis for many matrix algebra problems.

🧮 Key Formulas

(A')' = A

(kA)' = kA'

(A + B)' = A' + B'

(A - B)' = A' - B'

(AB)' = B'A'

det(A') = det(A)

trace(A') = trace(A)

If A is symmetric, A' = A

If A is skew-symmetric, A' = -A

Any square matrix A can be expressed as P + Q, where P = (A + A')/2 (symmetric) and Q = (A - A')/2 (skew-symmetric)

✅ Key Points for JEE

1The property (AB)' = B'A' is critical and frequently tested; remember to reverse the order of multiplication.
2Transpose operation distributes over matrix addition/subtraction and commutes with scalar multiplication: (A+B)' = A'+B' and (kA)' = kA'.
3Understanding the definitions and properties of symmetric (A'=A) and skew-symmetric (A'=-A) matrices is essential, especially for square matrices.
4Any square matrix can be uniquely decomposed into the sum of a symmetric and a skew-symmetric matrix; this decomposition is a common problem type.
5The determinant and trace of a matrix remain unchanged after transposition (det(A') = det(A) and trace(A') = trace(A)).

⚠️ Common Mistakes

✕Incorrectly applying (AB)' as A'B' instead of B'A'.
✕Confusing the conditions for symmetric and skew-symmetric matrices, or applying them to non-square matrices without careful consideration.
✕Errors in scalar multiplication within transpose operations, e.g., (kA)' = k'A' (k is a scalar, so k'=k).
✕Forgetting that the decomposition into symmetric and skew-symmetric components applies only to square matrices.

📝 Practice Questions

See all

Q22.Let M denote the set of all real matrices of order 3 × 3 and let S = {−3, −2, −1, 1, 2}. Let S1 = {A = [aij] ∈M : A = AT and aij ∈ S, ∀i, j}, S2 = {A = [aij] ∈M : A = −AT and aij ∈ S, ∀i, j}, S3 = {A = [aij] ∈M : a11 + a22 + a33 = 0 and aij ∈ S, ∀i, j}. If n ( S1 ∪2 US3) = 125α, then α equals _______

2025·NumericalHard

Q24.Let S = ∈Z : Am2 + Am = 3I , where A = 2 −1 . Then n(S) is equal to ______. {m −A−6} [ 1 0 ]

2025·NumericalMedium

Q4. If A, B , and (adj (A−1) + adj (B−1)) are non-singular matrices of same order, then the inverse of A(adj (A−1) + adj (B−1))−1 B , is equal to (1) AB−1 + A−1 B (2) adj (B−1) + adj (A−1) BA−1 (3) AB−1 (4) 1 (adj(B) + adj(A)) |A| + |B| |AB|

2025·MCQMedium

Q16.If I = ∫ 0π 3 dx, then ∫210 sin4x sinx+cos4x cos xx 2 2 x sin x+cos (1) π2 (2) π2 12 4 (3) π2 (4) π2 16 8 ∣∣ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2025 (23 Jan Shift 2) JEE Main Previous Year Paper

2025·MCQMedium

Q11.Let A = [aij] = [ log5log51288 log4log4255 ] . If Aij is the cofactor of aij, Cij = ∑2k=1 aikAjk, 1 ≤i, j ≤2, and C = [Cij], then 8|C| is equal to : (1) 288 (2) 222 (3) 242 (4) 262

2025·MCQMedium

Q5. Let A = [aij] be a matrix of order 3 × 3, with aij = (√2)i+j . If the sum of all the elements in the third row of A2 is α + β√2, α, β ∈Z, then α + β is equal to : (1) 280 (2) 224 (3) 210 (4) 168

2025·MCQMedium

NCERT Chapters

Class 12 Maths Ch 3: Matrices