(Reuters) - The U.S. Congress will try to establish who is to blame for at least 13 auto-related deaths over the past decade, as public hearings are launched on Tuesday on General Motors' slow response to defective ignition switches in cars.

Despite tougher laws being enacted in 2000 and 2010 to encourage automakers and the National Highway Traffic Safety Administration (NHTSA) to aggressively root out safety concerns, it took GM more than 10 years to acknowledge publicly that it had a potentially fatal problem.

Democrats on the House Energy and Commerce Committee, which is holding the first hearing, floated a new memo on Tuesday saying its analysis of documents found that GM still has not reported to federal regulators many cases involving ignition-switch concerns registered by consumers and GM technicians.

"At the same time GM was receiving these consumer complaints, the company continued to deny any defect," the House Democrats' memo said. The memo was referring to 133 cases, some dating from June 2003, with the "vast majority" unreported.

The memo noted that while automakers must provide a broad summary of warranty data to NHTSA, there currently is no requirement that they proactively submit warranty claims to the agency.

So far, GM has recalled 2.6 million cars to replace ignition switches that could unexpectedly stall out engines, prevent airbags from deploying and make power brakes and power steering inoperable.

Families of victims killed in crashes involving GM cars held an emotional meeting with GM Chief Executive Officer Mary Barra in the company's Washington offices on Monday night.

Laura Christian, whose daughter Amber Rose was killed in a 2005 Chevy Cobalt in Maryland, told Reuters after the meeting: "I pressed her (Barra) to consider whether she would take these cars off the road and she told me again how these cars were safe as long as you only had the one key (in the ignition with no other keys or fobs attached to a key-ring). I debated that."

Christian said "everybody was crying the entire time" during the meeting. "There were horrific stories."

Documents that GM and NHTSA turned over to the House committee have provided new insights into GM's practices.

They include decisions to install ignition switches in Chevrolet Cobalts, Saturn Ions and other models that did not meet all of the company's own specifications.

'LIVES ARE AT STAKE'

Some Republican aides on the committee have cautioned, however, that there were 60 specifications just for the ignition switch and it is not clear what the significance is of one specification being below-standard. The committee is sure to explore that question during the hearing, which is scheduled to start at 2 p.m.

Some in Congress are beginning to wonder whether more people died in cars outfitted with faulty switches, beyond the 13 GM identified, as they review documents pointing to a redesigned replacement part that also could be substandard.

"Lives are at stake, and we will follow the facts where they take us as we work to pinpoint where the system failed," House Energy and Commerce Committee Chairman Fred Upton, a Michigan Republican, said on Sunday.

Barra will be on the receiving end of questions by Upton and other members of the panel's Oversight and Investigations Subcommittee. She has repeatedly apologized for her company's performance.

Barra, who became CEO in January, says in prepared testimony released by the committee that she "cannot tell you why it took years for a safety defect to be announced." She promised to get to the root of the problem.

The congressional committee might not want to patiently wait, however, and it could call lower-level GM employees to testify at later hearings or even former CEOs.

NHTSA Acting Administrator David Friedman also will be on the hot seat on Tuesday, as many lawmakers are expected to ask why the regulatory agency was not more aggressive in identifying the problem and forcing GM to act.

For Barra and GM the stakes are high. The hearings, coupled with a Justice Department criminal investigation and a series of lawsuits, could further hurt the automaker's reputation among consumers and make for mounting legal fees.

Nonetheless, GM's stock was up 1.8 percent, or 63 cents, at $35.05 in morning trading on the New York Stock Exchange on Tuesday.

While the stock at one point dropped 10 percent over the course of a week in the aftermath of NHTSA opening its investigation, it has mostly recovered and is off only 1.4 percent since the first wave of ignition switch recalls was announced on February 13.

(Additional reporting by Julia Edwards; Writing by Richard Cowan; Editing by Karey Van Hall, Lisa Shumaker and Tom Brown)