Sophie Matisse in front of her great-grandfather's The Parakeet and the Mermaid. Photograph: Nils Jorgensen/Rex Features
There are several striking firsts at Tate Modern's new Matisse show: it is the first big exhibition in a generation to examine his vibrantly coloured paper cutouts, the first in the UK to display four important blue nudes together, and the first Tate show to be shown at the cinema.
It is also the first time the Tate will allow one its star attractions, its prized Matisse cutout The Snail, to leave the country. It will travel to New York in October – a measure of the importance of the exhibition, said the Tate director, Sir Nicholas Serota, one of the co-curators. "This kind of show just doesn't happen more than once in a lifetime," he said. "This is the largest show of this body of works and contains most of the major works."
The show explores the paper cutouts that Matisse, one of the greatest of all 20th-century artists, began making in his later life, between 1937-54. It brings together 130 gloriously coloured works, more than any before, and explores how and why Matisse began to make work in a new way so late in life.
The show, which opens to the public on Thursday, argues that the artist's techniques might look simple – paper, scissors, glue – but they were both "radical and groundbreaking".
The Matisse family have been closely involved in an exhibition that Serota said he had wanted to do for more than 30 years. Sophie Matisse, the artist's great-granddaughter and an artist herself, said: "It is such an amazing moment for me to see all these pieces together. It's just very moving … maybe I'll be able to talk about it better in a week when I've absorbed more."
She said she was particularly moved to see film footage of Matisse – old, wheelchair-bound, grumpy – at work. "To see him cutting … his scissors look like they just swam through the paper. To see his intent, his focus, it was like he'd done it a million times before. It's very beautiful and surprising."
Matisse was often ill or confined to bed during these years and was expected to die much sooner than he did, hanging on and defying medical predictions.
What emerges from the show is how vital Matisse was in his late years. He was "completely electric and alive and young," said Sophie. "He knew he only had another five minutes and he was going to make the best of it. You look at the works and he is not old at all."
Sophie, who has remarkable artistic forebears – her grandmother's second husband was Marcel Duchamp – recalled her family not talking about Matisse a great deal when she was young. "The presence was so intense, like a giant pink or red or purple elephant in the room, nobody really needs to talk about it, it's just there."
Serota said it had taken all the Tate's powers of persuasion to get galleries to lend what are often some of their star works.
The gallery is particularly pleased to be showing together works that are considered among the artist's most important: four blue nudes made in 1952. The works – two from the Pompidou, one from the Matisse Museum in Nice and one from the Beyeler Foundation in Basel – have been displayed together only a handful of times, and never in the UK.
There are some massive works in the show, not least Large Composition with Masks, from the National Gallery of Art in Washington, which is nearly 10 metres by 3.5 metres and is meant as a ceramic tile decoration for a Los Angeles couple's patio. The co-curator Nicholas Cullinan said: "Matisse got very excited and this was his first attempt to meet the commission, he just filled one wall of his studio and it is an incredible composition that harks back to Islamic tile decorations in the Alhambra."
The couple loved it but admitted it was three times too big. So he had another go, and another – "bear in mind it was the last year of his life," said Cullinan – and the fourth was accepted. "His appetite for work was really astonishing."
Tate is expecting big crowds. Serota said: "For many people [it] will be the most evocative and compelling show that London has ever seen."
For those who do not make it, Tate is collaborating with Seventh Art Productions – responsible for Leonardo Live from the National Gallery in 2011 – to bring the exhibition to cinema audiences.
Matisse Live on 3 June will include a live tour as well as interviews with experts, friends of the artist and archive footage of Matisse at work.
• Henri Matisse: The Cut-Outs is at Tate Modern from 17 April to 7 September, and at the Museum of Modern Art in New York from 14 October to 15 February.
Matisse exhibition at Tate Modern hailed by critics
The BBC's Helen Drew takes a look inside the exhibition
Critics have praised one of the largest collections of Henri Matisse's "cut-out" artworks ever assembled, for an exhibition opening at Tate Modern.
The French artist cut out paper shapes for collages when ill-health prevented him from painting, producing famous pieces such as The Snail and Blue Nude.
The Daily Telegraph said the London gallery must know it has a "winner" with its "outstanding" exhibition.
"I eat it with my eyes and never feel sated," said The Guardian's critic.
Many of the items will be seen together for the first time in the exhibition, which opens on Thursday and features about 130 artworks from the latter stage of Matisse's career.
Blue Nude  is a key example of the artist's skill with collage
The Telegraph said that "the joy of the cut-outs is their simplicity".
The paper's critic said the artworks were made from "modest materials" using "basic techniques" but that the artist "reduces art to the essentials of colour, shape and pattern".
"Yet precisely because they offer us instant visual gratification, it is easy to forget how innovative they actually are," he wrote.
The Guardian added that the show was "ravishing, filled with light and decoration, exuberance and a kind of violence" adding that it was "about more than just pleasure".
"Matisse created a universe that filled the room around him, spilling from the walls to the floor."
Matisse worked from a wheelchair after treatment for cancer and the exhibition compiles work dating from 1937 to 1954, when he died aged 84 of a heart attack.
Sir Nicholas Serota, the Tate director and co-curator of the show, told the BBC the works displayed great skill.
"Cut-out sounds a bit simplistic, they are very sophisticated objects.
"The brilliance is that he took the method of a child and deployed it with all the sophistication of an artist who had been painting for 60 years."
He said that the artworks were "incredibly influential" on a generation of American painters in the 60s and 70s.
"The colour is really intense, the colour is brilliant, it's really not quite what we associate with the immediate post-war years in Europe. He's really on his own."
The Economic Voice added that the exhibition "re-examines the cut-outs in terms of the methods and materials that Matisse used, and their double lives, first as contingent and mutable in the studio and ultimately as permanent works through mounting and framing".
The exhibition will be at the London gallery until 7 September before it travels to New York's Museum of Modern Art in mid-October. It can also be seen by cinema-goers from 3 June with the launch of Matisse Live.