 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/18/a/ii/solution

Expanding the determinant along the first row gives

det M = - a^{2} - b^{2} + 3 b + 3.

Under the matrix trace condition

a^{2} = 3

, this simplifies to

det M = b (3 - b)

. A square matrix is invertible exactly when its determinant is nonzero. Thus

a = \pm 3, b \in R ∖ {0, 3} .

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