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

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

The matrix trace is the sum of diagonal entries:

Tr M = a^{2} + 1 - 1 = a^{2} .

Consequently

a = \pm 3, b \in R arbitrary .

The first diagonal entry is

a^{2}

in the original PDF.

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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