Types — Symmetric, skew-symmetric, orthogonal
Matrices
6
JEE Qs
8%
Hard
45
min
Thoroughly understand the properties of transpose and matrix multiplication, as they are fundamental to identifying and manipulating symmetric, skew-symmetric, and orthogonal matrices in problem-solving.
🧮 Key Formulas
✅ Key Points for JEE
- 1For a symmetric matrix A, the elements satisfy a_ij = a_ji for all i, j. The diagonal elements can be any real number.
- 2For a skew-symmetric matrix A, the elements satisfy a_ij = -a_ji for all i, j. This implies that all diagonal elements (a_ii) must be zero.
- 3If A is an orthogonal matrix, its inverse A^(-1) is simply its transpose A^T. This significantly simplifies finding inverses.
- 4The sum/difference of two symmetric matrices is symmetric. The sum/difference of two skew-symmetric matrices is skew-symmetric.
- 5The determinant of an orthogonal matrix is always +1 or -1. This can be used as a quick check for orthogonality or to solve problems involving orthogonal matrices.
⚠️ Common Mistakes
- ✕Confusing the definitions of symmetric (A^T = A) and skew-symmetric (A^T = -A) matrices.
- ✕Incorrectly assuming diagonal elements of a symmetric matrix are zero (only true for skew-symmetric matrices).
- ✕Errors in matrix multiplication when verifying the orthogonality condition A^T A = I.
- ✕Forgetting that (AB)^T = B^T A^T when dealing with products of matrices and their transposes.
📝 Practice Questions
See allQ22.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 _______
Q24.Let S = ∈Z : Am2 + Am = 3I , where A = 2 −1 . Then n(S) is equal to ______. {m −A−6} [ 1 0 ]
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|
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
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
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
NCERT Chapters
- Class 12 Mathematics Ch 3: Matrices