Would an old head injury make someone to have a short temper?

Yes, but depends: This would depend on the nature, location, and severity of the old traumatic brain injury (tbi). Certainly if the tbi was more severe, one could have more long term-delayed sequelae from it. If the area of injury (as determined by ct or MRI scans) involved areas that controlled emotion, memory, speech, this certainly could lead to behavioral issues such as short temper.

BRAIN INJURY: If this was indeed a mild traumatic brain injury, not unique to have mood and behavioral symptoms associated, and indeed, many patients may be very sensitive and labile to emotional interactions, where they become irritable and over-react. Certainly, a number of head injury folks display an emotional dyscontrol syndrome, with a "pseudo-bulbar affect". A drug nuedexta may help.

Yes: More severe brain injuries can cause changes in personality and mood. It really depends on the seriousness of the injury. In general lasting problems are more common the longer the loss of consciousness and amnesia in the period of time directly before or after a head injury. If you are worried that you had a brain injury that has not been properly diagnosed see a neurologist immediately.