Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 18

The matrix multiplication $A\mathbf{c}$ adds each row. Equating it to $\mathbf{b}$ gives the linear systemThe last equation is the sum of the first two, so it supplies no further restriction. Subtracting the first from the second gives $2\beta =2$, hence $\beta =1$, and then $\gamma =1-\alpha$. Thus all parameter choices are

Substituting $\beta =1$ and $\gamma =1-\alpha$ into the matrix givesIts row sums are $4,4,6$, checking that $A_{\alpha }\mathbf{c}=\mathbf{b}$ for every real $\alpha$.

Expand the determinant along the first row:The cubic terms cancel in the expansion.

Completing the square in the determinant givesfor every real $\alpha$. A square matrix is invertible exactly when its determinant is nonzero, so $A_{\alpha }$ is nonsingular for every real $\alpha$.

Let $p(t)=\det (tI-A_{\alpha })$ be the real characteristic polynomial. It is a monic cubic, so $p(t)\to +\infty$ as $t\to +\infty$. At zero,The intermediate value theorem therefore gives some $t_0>0$ with $p(t_0)=0$. A root of the characteristic polynomial is an eigenvalue, hence $A_{\alpha }$ has a positive real eigenvalue for every real $\alpha$. This uses the general fact that an odd-dimensional real matrix with positive determinant has a positive eigenvalue.