Boston said the officer has "no criminal culpability" and called the use of force "justified" in the shooting.
In a Tuesday afternoon press conference, Boston said she met with Williams' family.
"It is no less a tragedy or heartbreaking," Boston said. "A life has been lost and for that we extend our deepest condolences to the family."
It happened around 6 p.m. on April 12, 2021, in the 2500 block of Terrace Trail. DeKalb County police said officers responded to a call about a man aggressively wielding a knife.
Investigators said the officers arrived and saw the man, who was still in possession of the knife. Police said the man lunged at officers with the knife, causing one of them to discharge their firearm.
The GBI handled the investigation.
"While this is not the resolution Mr. Williams family is seeking, it is the appropriate resolution under the law," Boston said.
Family demands justice for Matthew Zadok Williams
His family said was he having a mental health crisis and they wish the police would have instead gotten him help.
"We have had the displeasure of looking at over 10 hours of body camera footage and nothing in that footage shows that Sergeant Perry should have shot into my brother’s home," Hannah Williams told FOX 5 last July.
A doctor hired by Williams's family said he would likely be alive today if officers aided him after they shot him.
Body cam video shows Williams come at an officer with a knife in his hand, then police shot at him. The footage then shows Williams went back into the house and officers fired at him again.
In the footage, Sgt. Devon Perry is heard pleading with Williams to surrender.
"Please, sir. I’m begging you. I’m a Black man, you’re a Black man. You don’t have to die today. I don’t want you to die today," Perry can be heard in the video.
Williams' family said Perry violated police policies that day and believes he should be held accountable for his actions.