Let’s look at some identities of determinant

  1. Determinant of Identity matrix = 1
                det (I) = 1
    Where I is identity matrix of any order (1 × 1, 2 × 2, 3 × 3, … n × n)

  2. det (A T ) = det A
          |A T | = |A|

  3. |AB| = |A| |B|

  4. |A −1 | = 1/(|A|)

    Proof :

    AA −1 = I

    |AA −1 | = |I|

    |A| |A −1 | = −1

    |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

      We know that

                   A (adj A) = |A|I

      Taking determinant both sides

        |A (adj A) | = ||A|I |

We know that

  |kA| = k n |A|

| A || adj A | = | A | 𝑛 |I|

| A || adj A | = | A | 𝑛 × 1

| adj A | = |𝐴| 𝑛 / |𝐴|

|adj A | = | A | 𝑛−1

| adj A | = | A | n-1

          where n is the order of determinant

 

 

  1. Chapter 4 Class 12 Determinants
  2. Concept wise

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.