Check sibling questions

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

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

1.jpg

Other important points

Also, look at

 

Now learn Economics at Teachoo for Class 12


Transcript

Number multiplied to matrix and determinant Matrix If a number is multiplied to matrix, it is multiplied to each element of the matrix 2 [■8(9&2&1@5&−1&6@4&0&−2)] = [■8(2×9&2×2&2×1@2×5&2×(−1)&2×6@2×4&2×0&2×(−2))] Determinant If a number is multiplied to determinant, it is multiplied to either one row, or one column 2 |■8(9&2&1@5&−1&6@4&0&−2)| = |■8(2×9&2×2&2×1@5&−1&6@4&0&−2)| Or |■8(2×9&2&1@2×5&−1&6@2×4&0&−2)|

Davneet Singh's photo - Teacher, Engineer, Marketer

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.