After she fell down the cellar stairs and broke some vertebrae a few months ago, the 82-year-old Arlington widow knew she would not be able to spend much time this spring tending the garden she loves.

For help she turned to Operation Match, a new home-sharing program that brings together people who have extra space with people who are looking for housing. Operation Match located some prospective housemates for the woman, who chose to offer a rent-free room to a younger man in exchange for a sense of security and a little help around the house and yard.

A middle-aged teacher returning from an overseas job with his 11-year-old daughter and a newly acquired puppy wanted his daughter to go to Fairfax County schools and live in a house instead of an apartment. But he found the cost of housing prohibitive.

He also turned to Operation Match, and as a result of its efforts is renting two bedrooms in a spacious house owned by a woman with a 2-year-old, to whom his daughter has become "a second mother."

Throughout Northern Virginia, hundreds of people have been caught in the crunch to find affordable housing; at the same time, others have room to spare and could use a little extra income from rent, need help with tasks they would have to pay others to do, or simply want companionship. Operation Match tries to put the two together.

A one-year pilot program sponsored by the Department of Housing and Urban Development through the Metropolitan Washington Council of Governments, Operation Match has received nearly 600 applications from so-called "seekers" and "providers" since it opened in the last four to six months in Alexandria, Arlington County, Prince William County and Falls Church, which also covers that portion of Fairfax County through a Falls Church mailing address. The Fairfax County Board of Supervisors last week voted to begin a similar, county-wide program next year, and Loudoun County hopes to have its program going in a few months.

Anyone over 18, regardless of income, can participate in Operation Match, which does not charge for its services.

"Many of the seekers are people going through a midlife crisis," said Jean Rule, coordinator of the Greater Falls Church program. "They're often people who have separated or divorced. . . . Women are frequently the providers, because they've got the house and kids."

Men, she said, are more often the seekers, since they "may be paying child support and looking for very cheap housing."

But singles and people just starting their careers also are staples of the program, Rule said.

Of the 40 providers that have been listed for Fairfax County, 30 have been women; in Falls Church, 12 of the 20 providers have been women.

While some people seeking housing through Operation Match may be in the midst of a marital breakup, many are simply victims of the economy, said Elaine Bronez, who coordinates the Arlington program, which is financed by a $30,000 HUD grant.

"Much of this really has to do with the local depression and hard times," Bronez said. "A lot of the people are middle-class professionals who are out of work. Some of them are people who have been riffed; some are self-employed and trying to save money.

"Some of them are people who lived in apartments which turned condo or the rent went up again and they couldn't stand it. . . . They're not like 22-year-olds who are just trying to get together to share a house. . . . Many of them have had homes and furniture and security and now are trying to come to grips with the fact that they have to take something less than they've been used to."

Since Operation Match regulations specify that each person in a shared-housing situation must have a bedroom of his or her own, it is often difficult to place a parent with one or more children, program officials say. Such parents often are providers, however, and take in someone who agrees to care for the children while the parent works during the day.

Operation Match officials screen both house owners and those seeking housing. They check references and visit the house to be shared. They ask about personal preferences and habits--whether they would want to share a house with someone with children or with a smoker, for example. After the interviews, both "seekers" and "providers" are given the names of about three persons to contact. The names may come from another jurisdiction, since the programs coordinate their search efforts and make referrals from one area to another, spokeswomen said.

The two parties make their own detailed arrangements according to their needs, determining whether rent should be paid, how much and what services might be expected of each other.

"We're simply a clearinghouse and facilitator, we do not make the final agreements for both parties ," said Nancy Berg, the coordinator for Alexandria's program. "It's a complicated procedure, getting them matched. And then they don't always want each other."

In Alexandria, Operation Match has received 90 applications and made eight matches since it began in January Most applicants so far have been providers, people with houses looking for housemates, Berg said. Nearly 10 percent of them have been elderly persons, she said, and 63 of the 90 have been women heads of household: widows left with large houses they don't want to leave and divorced or separated women, many with small children.

The city received a HUD grant of $23,450, which is expected to be supplemented by $10,000 in city funds, to carry the program through June 1983, Berg said.

In neighboring Arlington, Bronez estimated, the program's 146 applicants since it opened in November have been divided about equally between providers and seekers, with women more numerous in both categories.

The Greater Falls Church program, which began in November, has a first-year budget of $28,000, representing almost equal HUD grants to the city and Fairfax County. The county turned its share over to Falls Church this year, a spokeswoman said, because the sum was considered too small to start an efficient program county-wide this year. Next year the county will spend $42,309 on the program.

In Prince William County, which received a $28,379 HUD grant, coordinator Theresa Rake Sidrow said there have been applications from 77 seekers and 134 providers since Operation Match began there in November.