medium difficulty8 min

Transpose of a Matrix

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

In one line: 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'.

Practice This Concept

Understand it

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.

Example

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

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let A = [[1, 0], [2, 3]] and B = [[4, 1], [5, -2]].

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

Method you can reuse

1Write entries of each column as a row.
2Change order from m x n to n x m.
3For sums, transpose each term.
4For products, reverse factor order and transpose each factor.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

A' has order 2 × 3.

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.

1
Enough for 1 mark
The transpose of a matrix A, denoted A' or A^T, is obtained by interchanging rows and columns of A.
2
Add for 2 marks
Example: If A = [[1, 2, 3], [4, 5, 6]], then A' = [[1, 4], [2, 5], [3, 6]].
3
Add for 3 marks
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.

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