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The 1st gen Matrix and 9th gen Corolla share a lot of the same mechanicals. The wheels are 5x100 bolt pattern, the OEM Matrix 16" wheels should be a 16x7, 45mm offset. The Corolla wheels will fit the Matrix, as I had a set of 2005 Corolla 15" steelies that I was going to use for snow tires on my old 2003 Matrix XRS.
The 2009 Matrix wheels will also fit, same bolt pattern, as long as it is not the 24L variants. Those use a slightly larger 5x114.3 bolt pattern, will not fit.
The 2003 Matrix tire on the 16" wheel is a 205/55R16. The 2005 Corolla on a 15" wheels is a 185/65R15 or a 195/65R15 depending on the trim level (CE, LE, S). Sizes are pretty close, should be within 1 MPH of actual speed at 60 MPH, in the case of the 185/65R15 - so speedo will not be off too much. The 195/65R15 are almost the same diameter as the 205/55R16 tires - so no issue there.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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