How to authenticate MIFARE 1K sectors with new authentication key ?

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C#

smartcard

Hello All,
Please let me know how can i authenticate the MIFARE sector with new Key as i write in sector trailer.

Suppose i have taken sector-8 block-35 and authenticate it with default key A.. FF FF FF FF FF FF so it allows me to read block 35 which gives me READ APDU.

FF B0 00 23 10
00 00 00 00 00 00 FF 07 80 69 FF FF FF FF FF FF

then i write new key in same sector trailer which gives me below APDU which is sucessfully write.

FF D6 00 23 10 A1 A2 A3 A4 A5 A6 FF 07 80 69 FF FF FF FF FF FF
90 00

again when i try to authenticate new sector 9 with key A it got authenticate well.

FF 86 00 00 05 01 00 27 60 01
90 00
Authentication success!

but now when i tried to authenticate block-35/sector 8 ... i got authentication failed error.

FF 86 00 00 05 01 00 23 60 01
63 00
Authentication failed!

What I have tried:

I have succesfully write the new key in sector trailer block 3. Here is the APDU for succesfully write key and i put the same access bit which i got on read command APDU.

FF D6 00 23 10 A1 A2 A3 A4 A5 A6 FF 07 80 69 FF FF FF FF FF FF
90 00