Matrices and Determinants - Formula Sheet and Summary

Let’s look at various properties of Matrices and Determinants

Addition and Subtraction of Matrices

• A + B = B + A
• (A + B) + C = A + (B + C)
• k (A + B) = kA + kB
  

Multiplication of matrices

• AB ≠ BA
• (AB) C = A (BC)
• Distributive law
  A (B + C) = AB + AC
  (A + B) C = AC + BC
• Multiplicative identity
  For a square matrix A
  AI = IA = A

Properties of transpose of matrix

• (A ^T ) ^T = A
• (kA) ^T = kA ^T
• (A + B) ^T = A ^T + B ^T
• (AB) ^T = B ^T A ^T

Symmetric and Skew Symmetric matrices

• Symmetric Matrix - If A ^T = A
  
  
• Skew - symmetric Matrix - If A ^T = A
  Note: In a skew matrix, diagonal elements are always 0 .
  
  
• For any square matrix A,
  (A + A ^T ) is a symmetric matrix
  (A − A ^T ) is a skew-symmetric matrix

Inverse of a matrix

For a square matrix A, if

      AB = BA = I

Then, B is the inverse of A

     i.e. B = A ⁻¹

We will find inverse of a matrix by

• Elementary transformation
• Using adjoint 

Properties of Inverse

1. For a matrix A,
   A ⁻¹ is unique, i.e., there is only one inverse of a matrix
   
   
2. (A ⁻¹ ) ⁻¹ = A
   
   
3. (𝑘 𝐴) ⁻¹ = 1/𝑘 𝐴 ⁻¹

   Note: This is different from
   (kA) ^T = k A ^T

4. (A ^-1 ) ^T = (A ^T ) ^-1
   
   
5. (A + B) ^-1 = A ^-1 + B ^-1
   
   
6. (𝐴𝐵) ⁻¹ = 𝐵 ⁻¹ 𝐴 ⁻¹

Important things to note in Determinants

1. Determinant of Identity matrix = 1
   det (I) = 1
   
   
2. |A ^T | = |A|
   
   
3. |AB| = |A| |B|
   
   
4. |A ⁻¹ | = 1/|𝐴|
   
   
5. |kA| = k ⁿ |A| where n is order of matrix
   
   
6. Similarly,
   |−A| = |−1 × A|
          = (−1) ⁿ × |A|
   
   
7. (adj A) A = A (adj) = |A|I
   
   
8. Deteminant of adj A
   |adj A| = |A| ^𝑛−1
   where n is the order of determinant

Number multiplied to matrix and determinant

Other important points

Also, look at

• Finding Inverse using Elementary Transformation
• Finding Inverse using Adjoint
• Properties of Determinant
• Difference between Matrices and Determinants