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

 

  1. Chapter 4 Class 12 Determinants
  2. Serial order wise

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)|

About the Author

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.