Question

A and B are square matrices such that A^2020=0 and AB=A+B then det(B)​

Answer 1

Answer:

Multiply no both sides by A^2019

Step-by-step explanation:

Answer 2

The det(B) of the given square matrices is 0

Step-by-step explanation:

Given as :

$A^{2020}$  = 0              .............1

A B = A + B          .............2

Let the det(B) =  $\left | B \right |$

According to question

From eq 2

A B = A + B  

multiplying both side by $A^{2019}$

i.e  $A^{2019}$ ( A B) = $A^{2019}$ (A + B)

Or,  $A^{2019}$ . A  .B = $A^{2019}$ . A + $A^{2019}$ . B

Or, $A^{2019+1}$ . B = $A^{2019+1}$ + $A^{2019}$ . B

Or, $A^{2020}$ .B = $A^{2020}$ + $A^{2019}$ . B

Now, From eq 1

∵   $A^{2020}$  = 0

So,  0 . B  = 0 + $A^{2019}$ . B

Or,  0 = 0 + $A^{2019}$ . B

∴     $A^{2019}$ . B = 0

or, B  = 0

det(B) = $\left | B \right |$   =  $\left | 0 \right |$

i.e   , det(B)  = 0

So, The det(B) of the given square matrices = $\left | B \right |$   =  0

Hence, The det(B) of the given square matrices is 0 Answer