You have a natural refactoring, lets call it cell_wrap,

sub cell_wrap {
    join '', map {"<tr><td>$_</td></tr>"} @_;
}
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Since you're just choosing between values in your conditional, the trinary op ?: comes to mind. Finally, I'd keep the foreach loop pretty much as you have it, though I'll pare it down to for and name no temporaries.

for (@$matrix_ref) {
    $html_str .= cell_wrap(
        $_->[0] eq $key
        ? $_->[2]
        : (@{$_}[0..2]) );
    $key = $_->[0];
}
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It would be a bigger improvement to separate logic from presentation by merging the records and then feeding $matrix_ref to a template engine like Html::Template or TT2.

Update: Arrgh! You changed your presentation code! Take this as another argument for templating.