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

 

 

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.