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Transpose of a Matrix

The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A.

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Student-friendly explanation

If A = [a_ij] has order m x n, then A' = [a_ji] has order n x m. This means the entry originally in row i and column j moves to row j and column i, so transpose changes positions but not the numerical values or signs. It works entry-wise over addition and scalar multiplication, but for a product it reverses the factor order: the transpose of AB is B'A', not A'B'. This reversal is important because matrix multiplication depends on order and AB and BA may be different or one of them may be undefined.

How to write this in exams

  1. 1

    Start with the exact idea

    The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A.

  2. 2

    Then show how to use it

    Write entries of each column as a row. Change order from m x n to n x m. For sums, transpose each term. For products, reverse factor order and transpose each factor.

  3. 3

    Add one concrete example

    If A = [[1, 2, 3], [4, 5, 6]], then A' = [[1, 4], [2, 5], [3, 6]].

  4. 4

    Avoid this incomplete answer

    Changing signs while taking transpose; transpose changes positions only, not signs.

Definition

The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A.

Example

If A = [[1, 2, 3], [4, 5, 6]], then A' = [[1, 4], [2, 5], [3, 6]].

Rule to remember

Key properties: (A')' = A, (A+B)' = A' + B' when A and B have the same order, (kA)' = kA', and (AB)' = B'A' when AB is defined.

Memory hook

For transpose, read every column of A as a row of A'. For a product, remember that the last factor comes first after transpose: (AB)' = B'A'.

Examples and method

Worked example

Let A = [[1, 0], [2, 3]] and B = [[4, 1], [5, -2]]. AB = [[4, 1], [23, -4]]. Therefore (AB)' = [[4, 23], [1, -4]]. Also B'A' = [[4, 5], [1, -2]][[1, 2], [0, 3]] = [[4, 23], [1, -4]]. Hence (AB)' = B'A'.

Method to apply

Write entries of each column as a row. Change order from m x n to n x m. For sums, transpose each term. For products, reverse factor order and transpose each factor.

Diagram support

No separate diagram is required; interchanging rows and columns can be shown directly by rewriting the matrix.

How CBSE asks it

Questions may ask for a direct transpose, verification of transpose properties, or use of transpose in symmetric and skew-symmetric matrices.

Avoid common mistakes

Common confusion

The most common error is writing (AB)' = A'B'. The correct order is reversed.

Common wrong answer

Changing signs while taking transpose; transpose changes positions only, not signs.

Exam tip

When transposing a product, reverse the order of factors before transposing each factor.

Quick check

If A is of order 3 x 2, what is the order of A'?

A' has order 2 x 3.

Answer writing and exam use

1-mark answer

The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A.

2-mark answer

The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A. Key properties: (A')' = A, (A+B)' = A' + B' when A and B have the same order, (kA)' = kA', and (AB)' = B'A' when AB is defined. If A = [[1, 2, 3], [4, 5, 6]], then A' = [[1, 4], [2, 5], [3, 6]].

3-mark answer

If A = [a_ij] has order m x n, then A' = [a_ji] has order n x m. This means the entry originally in row i and column j moves to row j and column i, so transpose changes positions but not the numerical values or signs. It works entry-wise over addition and scalar multiplication, but for a product it reverses the factor order: the transpose of AB is B'A', not A'B'. This reversal is important because matrix multiplication depends on order and AB and BA may be different or one of them may be undefined. Key properties: (A')' = A, (A+B)' = A' + B' when A and B have the same order, (kA)' = kA', and (AB)' = B'A' when AB is defined. Let A = [[1, 0], [2, 3]] and B = [[4, 1], [5, -2]]. AB = [[4, 1], [23, -4]]. Therefore (AB)' = [[4, 23], [1, -4]]. Also B'A' = [[4, 5], [1, -2]][[1, 2], [0, 3]] = [[4, 23], [1, -4]]. Hence (AB)' = B'A'. Questions may ask for a direct transpose, verification of transpose properties, or use of transpose in symmetric and skew-symmetric matrices. Changing signs while taking transpose; transpose changes positions only, not signs.
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