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CBSE · Class 12 · Mathematics

Transpose, Symmetric and Skew Symmetric Matrices (Ch 3: 3.5–3.6)

2 concepts · 16 practice questions with answers · Free to read

Transpose of a matrix

From daily life

Turn a class timetable so that days become columns and periods become rows. The same information appears, just turned on its side.

Simple definition

The transpose A′ of a matrix A is obtained by writing the rows of A as columns. If A is m × n, then A′ is n × m. Properties: (A′)′ = A, (kA)′ = kA′, (A + B)′ = A′ + B′, (AB)′ = B′A′.

Formula: (AB)′ = B′ A′ (order reverses)

Rule 1: Transpose a matrix — Row 1 becomes column 1, and so on.

  • Easy: [1 2 3 ; 4 5 6]′ = [1 4 ; 2 5 ; 3 6].
  • Medium: A 2 × 3 matrix has a 3 × 2 transpose.
  • Exam-type: For A = [1 2 ; 3 4], B = [0 1 ; 1 0], (AB)′ = [2 4 ; 1 3] = B′A′.

Rule 2: Use the properties — Apply them before multiplying out.

  • Easy: (A′)′ = A for every matrix.
  • Medium: (A + B)′ = A′ + B′.
  • Exam-type: (ABC)′ = C′B′A′, reversing all the way.
Remember

The transpose of a product reverses the order of the factors.

Common mistake

Writing (AB)′ = A′B′.

Memory clue

Socks and shoes again: transpose reverses the order.

Practice questions

Easy
  1. 1Find the transpose of [2 5 ; 7 1].
  2. 2A is 4 × 3. What is the order of A′?
  3. 3Find the transpose of [1 ; 2 ; 3].
Medium
  1. 4If A = [1 2 ; 3 4] and B = [5 6 ; 7 8], find (A + B)′.
  2. 5Find (3A)′ for A = [1 0 ; 2 5].
  3. 6Verify (AB)′ = B′A′ for A = [1 2] and B = [3 ; 4].
Challenging
  1. 7If A′ = [1 4 ; 2 5 ; 3 6], find A.
  2. 8What is (A′)′ for any matrix A?
Show answers (try them yourself first)
  1. 1[2 7 ; 5 1]
  2. 23 × 4
  3. 3[1 2 3]
  4. 4A + B = [6 8 ; 10 12], so (A + B)′ = [6 10 ; 8 12]
  5. 53A = [3 0 ; 6 15], so (3A)′ = [3 6 ; 0 15]
  6. 6AB = [11], (AB)′ = [11]; B′A′ = [3 4] [1 ; 2] = [11]
  7. 7A = [1 2 3 ; 4 5 6]
  8. 8A itself

Symmetric and skew symmetric matrices

From daily life

A distance table between cities is the same read across or down, because the distance from A to B equals the distance from B to A. That is a symmetric matrix.

Simple definition

A square matrix A is symmetric if A′ = A (that is aᵢⱼ = aⱼᵢ) and skew symmetric if A′ = −A (that is aᵢⱼ = −aⱼᵢ). The diagonal entries of a skew symmetric matrix are all 0. Every square matrix is the sum of a symmetric matrix and a skew symmetric matrix.

Formula: A = ½ (A + A′) + ½ (A − A′)

Rule 1: Classify — Compare each entry with its mirror image across the diagonal.

  • Easy: [1 2 ; 2 3] is symmetric.
  • Medium: [0 5 ; −5 0] is skew symmetric.
  • Exam-type: [2 −1 4 ; −1 0 3 ; 4 3 5] is symmetric.

Rule 2: Split into symmetric and skew parts — Use the formula.

  • Easy: For A = [1 3 ; 5 7], P = ½ (A + A′) = [1 4 ; 4 7].
  • Medium: Q = ½ (A − A′) = [0 −1 ; 1 0].
  • Exam-type: P + Q = [1 3 ; 5 7] = A, as it should be.
Remember

A + A′ is always symmetric and A − A′ is always skew symmetric.

Common mistake

Forgetting that the diagonal of a skew symmetric matrix must be zero.

Memory clue

Symmetric = mirror image; skew = mirror image with the sign flipped.

Practice questions

Easy
  1. 1Is [3 4 ; 4 9] symmetric?
  2. 2Is [0 −2 ; 2 0] skew symmetric?
  3. 3What are the diagonal entries of a skew symmetric matrix?
Medium
  1. 4Find x if [2 x ; 5 1] is symmetric.
  2. 5Find a if [0 a ; −3 0] is skew symmetric.
  3. 6Express [2 6 ; 4 8] as the sum of a symmetric and a skew symmetric matrix.
Challenging
  1. 7Show that A + A′ is symmetric for any square matrix A.
  2. 8Is the product of two symmetric matrices always symmetric? Test A = [1 2 ; 2 0] and B = [0 1 ; 1 3].
Show answers (try them yourself first)
  1. 1Yes
  2. 2Yes, because its transpose is [0 2 ; −2 0] which is −A
  3. 3All 0
  4. 4x = 5
  5. 5a = 3 because a₁₂ = −a₂₁ = 3
  6. 6Symmetric part [2 5 ; 5 8] plus skew part [0 1 ; −1 0]
  7. 7(A + A′)′ = A′ + A = A + A′
  8. 8No: AB = [2 7 ; 0 2], which is not symmetric