Andy Burnham has backed mandatory suicide prevention training for social workers following the "devastating" death of Matilda Davis, a parent-carer who killed herself minutes after meeting professionals. The Guardian revealed in August how the "absolutely inspirational" 36-year-old took her life after feeling "completely exhausted" and failed by England's social care system.

Davis, a former art teacher, had told multiple professionals she was struggling to cope with the significant care for her seven-year-old son and was "paralysed with anxiety" about her looming divorce from her estranged husband. She died on 3 October last year, minutes after an emotional meeting with social workers who had not received suicide prevention training.

Burnham was asked about Davis's case at prime minister's questions on Wednesday and said he supported the extension of this training to all local authorities in England.

Manuela Perteghella, the Liberal Democrat MP for Stratford-upon-Avon, where Davis lived, said: "My constituent Matilda Davis was a wonderful and devoted mum and a full-time carer for her disabled seven-year-old child. Despite providing round the clock care, she received almost no support leaving her physically and emotionally exhausted. Last October, social workers visited Matilda and she took her own life 15 minutes after they left. She was 36."

Perteghella highlighted University of Birmingham research showing that 41% of parent-carers have contemplated suicide, and asked Burnham to meet her and Matilda's family to discuss "mandatory suicide prevention training for local authorities and other measures so that no more normal families suffer such a devastating loss". Burnham replied: "I will ensure that a minister meets the honourable lady to discuss that proposal because I find myself in agreement with it."

The prime minister praised the Lib Dem leader, Ed Davey, for his work campaigning on behalf of unpaid carers, whose failure by the state has been brought into light by an award-winning Guardian investigation into the carer's allowance scandal. Burnham said these carers, who often gave up work and other commitments while saving the government billions of pounds to look after severely disabled loved ones, could often "sink under their responsibilities and find it very, very difficult". He added: "Sadly in the case of Matilda, [Perteghella's] constituent, obviously a devastating situation. I will arrange that meeting and I thank [Perteghella] for raising it in the House [of Commons] today."

At Davis's inquest in March, social workers admitted they did not ask her whether she had thoughts of suicide despite knowing she was in significant distress, nor was she directed to crisis organisations. As a result, the coroner, Deborah Sewell, issued a rare prevention of future deaths notice ordering Warwickshire council to make suicide prevention training mandatory for 600 members of staff, primarily social workers.

Davis's brother, Joel Harley-Davis, said the family was "really reassured" by Burnham's response and that his sister would have been "proud" to make a positive difference. "We all believe that this training would have been enough to save Matilda's life," he said. "If those social workers had asked her [whether she was considering ending her life] Matilda was so honest she would have said yes. This needs to be made mandatory - you can't leave it up to councils to decide whether to put it in place because it creates a postcode lottery. It doesn't need to be this really expensive thing. All the literature and advice says that if in doubt about whether someone is suicidal, ask that direct difficult question."

Guidance issued by the mental health charity Mind states that it is best practice not to "skirt around the topic of suicide" and instead ask direct questions such as: "Are you having suicidal thoughts?" when speaking to someone who might want to kill themselves.

Harley-Davis, 30, said his sister felt "failed and let down" by England's social care system and was passionate about advocating on behalf of other parent-carers. He added: "Matilda was an incredibly stoic individual but that's not to say she didn't need someone to offer that respite. Matilda needed that professional support and a degree of empathy and I don't think she got that, especially on the day she died."

In the UK and Ireland, Samaritans can be contacted on freephone 116 123. In the US, you can call or text the National Suicide Prevention Lifeline on 988, chat on 988lifeline.org, or text HOME to 741741 to connect with a crisis counsellor. In Australia, the crisis support service Lifeline is 13 11 14. Other international helplines can be found at befrienders.org