Paper 1, Section II, A

Vectors and Matrices | Part IA, 2018

What is the definition of an orthogonal matrix MM ?

Write down a 2×22 \times 2 matrix RR representing the rotation of a 2-dimensional vector (x,y)(x, y) by an angle θ\theta around the origin. Show that RR is indeed orthogonal.

Take a matrix

A=(abbc)A=\left(\begin{array}{ll} a & b \\ b & c \end{array}\right)

where a,b,ca, b, c are real. Suppose that the 2×22 \times 2 matrix B=RARTB=R A R^{T} is diagonal. Determine all possible values of θ\theta.

Show that the diagonal entries of BB are the eigenvalues of AA and express them in terms of the determinant and trace of AA.

Using the above results, or otherwise, find the elements of the matrix

(1221)2N\left(\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right)^{2 N}

as a function of NN, where NN is a natural number.

Typos? Please submit corrections to this page on GitHub.