Types of Matrix

Types of Matrices — Definitions & Examples

A concise reference listing common matrix types with a short definition and two examples for each. Perfect for students, teachers, and blog readers.

Table of contents

• Row Matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Zero (Null) Matrix
• Identity Matrix
• Diagonal Matrix
• Scalar Matrix
• Triangular Matrices
• Symmetric Matrix
• Skew-Symmetric Matrix
• Singular Matrix
• Non-Singular Matrix
• Orthogonal Matrix
• Sparse Matrix

Row Matrix

Definition: A matrix with only one row.

Examples

[ 3  5  7 ]
[ -2  4  9  1 ]

Column Matrix

Definition: A matrix with only one column.

Examples

[ 4
 -1
  2 ]

[ 7
  0 ]

Square Matrix

Definition: A matrix with an equal number of rows and columns (n × n).

Examples

[ 1  2
  3  4 ]

[ 5  6  7
  1  0  2
  3  4  8 ]

Rectangular Matrix

Definition: A matrix where the number of rows and columns are not equal.

Examples

[ 1  2  3
  4  5  6 ]

[ 7  8
  9 10
 11 12 ]

Zero (Null) Matrix

Definition: A matrix in which every element is zero.

Examples

[ 0  0
  0  0 ]

[ 0  0  0 ]

Identity Matrix

Definition: A square matrix with 1's on the main diagonal and 0's elsewhere (I_n).

Examples

I2 = [ 1  0
       0  1 ]

I3 = [ 1  0  0
       0  1  0
       0  0  1 ]

Diagonal Matrix

Definition: A square matrix where all non-diagonal entries are zero (only diagonal may be non-zero).

Examples

[ 3  0  0
  0  5  0
  0  0  7 ]

[ 1  0
  0 -4 ]

Scalar Matrix

Definition: A diagonal matrix in which every diagonal element is the same scalar value.

Examples

[ 5  0
  0  5 ]

[ -3  0  0
   0 -3  0
   0  0 -3 ]

Triangular Matrices

Upper triangular: all entries below the main diagonal are zero. Lower triangular: all entries above the main diagonal are zero.

Upper triangular examples

[ 1  2  3
  0  5  6
  0  0  7 ]

[ 4 -2
  0  9 ]

Lower triangular examples

[ 2  0  0
  3  4  0
  5  6  7 ]

[ 1  0
 -3  8 ]

Symmetric Matrix

Definition: A square matrix equal to its transpose (A = A^T).

Examples

[ 2  3
  3  5 ]

[ 1  4  7
  4  2  6
  7  6  3 ]

Skew-Symmetric Matrix

Definition: A square matrix where A^T = -A. Diagonal elements must be zero.

Examples

[  0 -2
   2  0 ]

[  0  3 -1
  -3  0  5
   1 -5  0 ]

Singular Matrix

Definition: A square matrix with determinant equal to zero (det(A) = 0).

Examples

[ 1  2
  2  4 ]  

[ 3  6
  1  2 ]

Non-Singular Matrix

Definition: A square matrix with non-zero determinant (det(A) ≠ 0). It has an inverse.

Examples

[ 1  2
  3  4 ]

[ 2  5
  1  3 ]

Orthogonal Matrix

Definition: A square matrix A such that A^TA = I (transpose is inverse).

Examples

[ 1  0
  0  1 ]

[ 0  1
 -1  0 ]

Sparse Matrix

Definition: A matrix with mostly zero entries (only a few non-zero elements).

Examples

[ 0  0  5
  0  0  0
  2  0  0 ]

[ 0  3
  0  0
  0  0 ]

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Integration by Part

 

Integration by Part

If u and v are two functions of x, then

Remark: The proper choice of u and v is very important to apply integration by part. We can choose the first function (u) as the function which comes first in the word ILATE. Here

I stands for the inverse trigonometric function (Ex. )

L stands for logarithmic function (Ex. )

A stands for the algebraic functions (Ex. ,  etc.)

T stands for the trigonometric functions (Ex. )

E stands for the exponential functions. (Ex. )

Example 1: Evaluate

Solution: Here as per the above remark ILATE  (Algebraic function) comes first and  (trigonometric function).

Thus let  and .

Now by formula

Example 2: Evaluate  

Solution: Here as per the above remark ILATE  (Algebraic function) comes first and  (trigonometric function).

Thus let  and .

Now by formula

Where   and

Here in second part of integration () we will apply integration by part again.

Here as per the above remark ILATE  (Algebraic function) comes first and  (trigonometric function).

Thus let  and .

Now by formula

Thus

Example 3: Evaluate

Solution: Here as per the above remark ILATE  (logarithmic function) comes first and  (Algebraic function).

Thus let  and .

Now by formula

Example 4: Evaluate

Solution: Given

let

Putting these values in (1), we get

Now, as per the above remark ILATE  (Algebraic function) comes first and  (trigonometric function).

Thus let  and .

Now by formula

Example 5: Evaluate

Solution: Given

Here the formula ILATE does not works.

We know that  and

Thus let  and .

Now by formula

Example 6: Evaluate

Solution: As per the remark ILATE  (inverse trigonometric function) comes first and  (Algebraic function).

Thus let  and .

Now by formula

 

Example 7: Evaluate

Solution: As per the remark ILATE  (logarithmic function) comes first and  (Algebraic function).

Thus let   and .

Now by formula

Example 8: Evaluate

Solution: let , we get

As per the remark ILATE  (inverse trigonometric function) comes first and  (Algebraic function). Thus let    and .

Now by formula

Example 9: Evaluate

Solution: let

Where