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Column of a Matrix

Column of a Matrix

A vertical set of numbers in a matrix.

3x3 matrix with second column [2, 5, 8] highlighted, illustrating a column of a matrix.

 

 

See also

Row of a matrix, element of a matrix

Key Formula

A=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]⇒Column j=[a1ja2j⋮amj]A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \quad \Rightarrow \quad \text{Column } j = \begin{bmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{bmatrix}
Where:
  • AA = An m × n matrix (m rows and n columns)
  • aija_{ij} = The element in row i and column j of the matrix
  • jj = The column number, ranging from 1 to n
  • mm = The total number of rows in the matrix
  • nn = The total number of columns in the matrix

Worked Example

Problem: Given the matrix B, identify the second column and state its entries.
Step 1: Write out the matrix B.
B=[538271469]B = \begin{bmatrix} 5 & 3 & 8 \\ 2 & 7 & 1 \\ 4 & 6 & 9 \end{bmatrix}
Step 2: Identify column 2 by reading down the second vertical position (j = 2) from top to bottom. The entries are the elements where the column index is 2: b₁₂, b₂₂, and b₃₂.
b12=3,b22=7,b32=6b_{12} = 3, \quad b_{22} = 7, \quad b_{32} = 6
Step 3: Write the second column as a column vector.
Column 2=[376]\text{Column 2} = \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix}
Answer: The second column of B is the column vector [3, 7, 6]ᵀ.

Another Example

This example differs by involving a non-square matrix and extracting all columns rather than just one, reinforcing how the matrix dimensions determine the number and length of columns.

Problem: A matrix C has dimensions 2 × 4. List all of its columns.
Step 1: Write out the matrix C.
C=[10423578]C = \begin{bmatrix} 1 & 0 & 4 & 2 \\ 3 & 5 & 7 & 8 \end{bmatrix}
Step 2: Since C is a 2 × 4 matrix, it has 4 columns. Each column has 2 entries (one from each row).
Step 3: Extract each column by reading downward at each position.
Col 1=[13],Col 2=[05],Col 3=[47],Col 4=[28]\text{Col 1} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \quad \text{Col 2} = \begin{bmatrix} 0 \\ 5 \end{bmatrix}, \quad \text{Col 3} = \begin{bmatrix} 4 \\ 7 \end{bmatrix}, \quad \text{Col 4} = \begin{bmatrix} 2 \\ 8 \end{bmatrix}
Step 4: Notice that the number of columns (4) equals the second dimension of the matrix size (2 × 4), and each column vector has a length equal to the number of rows (2).
Answer: Matrix C has four columns: [1, 3]ᵀ, [0, 5]ᵀ, [4, 7]ᵀ, and [2, 8]ᵀ.

Frequently Asked Questions

What is the difference between a row and a column of a matrix?
A row is a horizontal set of numbers running left to right across the matrix, while a column is a vertical set of numbers running top to bottom. In the notation a_{ij}, the first subscript i identifies the row and the second subscript j identifies the column. A matrix with m rows and n columns has m row vectors and n column vectors.
How do you find the number of columns in a matrix?
The number of columns equals the second number in the matrix's dimensions. If a matrix is described as m × n, then n is the number of columns. For example, a 3 × 5 matrix has 5 columns, and each column contains 3 entries.
What is a column vector?
A column vector is a matrix that has only one column. It has dimensions m × 1, where m is the number of entries. Every column of a larger matrix can be treated as a column vector on its own. Column vectors are fundamental in linear algebra for representing points, directions, and solutions to systems of equations.

Column of a Matrix vs. Row of a Matrix

Column of a MatrixRow of a Matrix
DirectionVertical (top to bottom)Horizontal (left to right)
NotationFixed column index j; varies row index iFixed row index i; varies column index j
Count in an m × n matrixn columnsm rows
Length of each vectorm entries per columnn entries per row
As a standalone vectorColumn vector (m × 1)Row vector (1 × n)

Why It Matters

Columns appear constantly in linear algebra: when you multiply a matrix by a vector, the result is a combination of the matrix's columns. Understanding columns is also essential for matrix operations like finding determinants, performing Gaussian elimination, and computing matrix–matrix products. In data science and statistics, each column of a data matrix typically represents a different variable or feature, so recognizing columns is crucial for interpreting real-world datasets.

Common Mistakes

Mistake: Confusing the row index and the column index in double-subscript notation. For example, reading a_{23} as "column 2, row 3" instead of "row 2, column 3."
Correction: Always remember the convention: the first subscript is the row, the second is the column. So a_{23} is in row 2, column 3. A helpful mnemonic: think "RC" — Row first, Column second — like the abbreviation for a remote control.
Mistake: Thinking the number of columns equals the first dimension. For instance, calling a 4 × 3 matrix one that has 4 columns.
Correction: The first number in m × n gives the rows, and the second gives the columns. A 4 × 3 matrix has 4 rows and 3 columns, not the other way around.

Related Terms