 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/1/a/ii/solution

When

A

B = A^{- 1} (A B)

. Apply i first to

A B

and then to

A^{- 1} (A B)

rowspace (A B) \subseteq rowspace (B) = rowspace (A^{- 1} A B) \subseteq rowspace (A B) .

Thus the two row spaces are equal. In particular, invertible left matrix multiplication preserves row rank.

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