Important things to note in Determinants

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