Column Matrix

A column matrix is a type of matrix that has only one column and multiple rows. It is represented as:

A=[a1a2a3an]

where A is an n×1 matrix, meaning it has n rows and 1 column.

Properties of a Column Matrix

1 Order of a Column Matrix

A column matrix is always of the order n×1, where n represents the number of rows.

Example

Consider a 4×1 column matrix:

A=[5273]

Here, the matrix has 4 rows and 1 column, so its order is 4×1.

2 Addition of Column Matrices

Two column matrices of the same order can be added by adding their corresponding elements.

Example

Let:

A=[246],B=[135]

Then, the sum is:

A+B=[2+14+(3)6+5]=[3111]

3 Scalar Multiplication

A column matrix can be multiplied by a scalar (a single number) by multiplying each element by the scalar.

Example

Let:

A=[314],k=2

Then, the scalar multiplication is:

kA=2×[314]=[628]

4 Transpose of a Column Matrix

The transpose of a column matrix converts it into a row matrix.

Example

Let:

A=[785]

Then, its transpose is:

AT=[785]

5 Multiplication of Two Column Matrices

Two column matrices cannot be directly multiplied unless one is transposed to form a row matrix.

Example

Let:

A=[23],B=[14]

To perform matrix multiplication, we take the transpose of A:

ATB=[23]×[14]

The result is:

ATB=(2×1)+(3×4)=2+12=10

6 Identity Property

Multiplying a column matrix by an identity matrix of compatible order results in the original matrix.

Example

Let:

I=[100010001],A=[523]

Then:

IA=[100010001]×[523]=[523]

7 Zero Column Matrix

A column matrix in which all elements are zero is called a zero column matrix.

Example

Z=[000]

For any column matrix A, adding Z does not change A:

A+Z=A