Q.Transpose of a column matrix is a column matrix.

The key idea is that the transpose of a column matrix (a matrix with a single column) is a row matrix, not a column matrix.

• A column matrix of size $m \times 1$ has $m$ rows and 1 column.
• The transpose operation swaps rows and columns. So the transpose of an $m \times 1$ matrix becomes a $1 \times m$ matrix.
• A $1 \times m$ matrix has one row and $m$ columns — that is a row matrix, not a column matrix. …

Why This Matters: The Core Idea of Transpose

The transpose operation flips a matrix over its main diagonal — rows become columns, and columns become rows. This is a fundamental transformation in linear algebra, and getting it right is essential for matrix multiplication, solving systems, and understanding vector spaces.

A column matrix (also called a column vector) has only one column and any number of rows. For example:

has dimensions $3 \times 1$ (3 rows, 1 column).

When you transpose it, every element that was in row $i$, column 1 moves to row 1, column $i$. So the result has 1 row and as many columns as the original had rows — that's a row matrix.

Step-by-Step Reasoning

1. Recall the definition of transpose.

   If $A$ is an $m \times n$ matrix, its transpose $A^T$ is an $n \times m$ matrix where $(A^T)_{ij} = A_{ji}$. In plain language: the first row of $A^T$ is the first column of $A$, the second row of $A^T$ is the second column of $A$, and so on.

2. Apply this to a column matrix.

   Let $C$ be a column matrix of size $m \times 1$:

Here $m$ is the number of rows, and there is exactly 1 column.

3. Perform the transpose. The first (and only) column of $C$ becomes the first row of $C^T$. So:

This is a $1 \times m$ matrix — exactly one row and $m$ columns.

4. Check the dimensions. Original: $m \times 1$ → Transpose: $1 \times m$. A $1 \times m$ matrix is called a row matrix (or row vector), not a column matrix. …

Method: Determining the Order (Shape) of a Transposed Matrix

Use this for any statement about what kind of matrix results from transposing. The transpose swaps the roles of rows and columns, so track the order carefully.

Steps

Step 1: Write the definition of transpose in terms of order.

If $A$ has order $m\times n$, then $A'$ (or $A^{T}$) has order $n\times m$, with $(A')_{ij}=a_{ji}$. Rows become columns and columns become rows.

Step 2: Substitute the given shape.

Express the described matrix as an order. A column matrix has one column, so it is $m\times 1$. Its transpose is therefore $1\times m$. …

Common Mistakes

Mistake 1: Thinking the shape is unchanged by transposing.

Why it's wrong: transpose swaps rows and columns, so an $m\times 1$ column becomes a $1\times m$ row. Correct approach: always convert the shape to its order and swap the two numbers.

Mistake 2: Confusing a column matrix with a row matrix by name. …