Answer to Question #282043 in Linear Algebra for Peyt

Question

$$16\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{matrix} a & b \\ c & d \end{matrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{matrix} a & b \\ c & d \end{matrix}$$

Answer 1

Solution:

$16\begin{pmatrix}a&b\\ \:c&d\end{pmatrix}\begin{pmatrix}a&b\\ \:c&d\end{pmatrix}\begin{pmatrix}a&b\\ \:c&d\end{pmatrix}\begin{pmatrix}a&b\\ \:c&d\end{pmatrix} \\=16\begin{pmatrix}a^2+bc&ab+bd\\ ca+dc&bc+d^2\end{pmatrix}\begin{pmatrix}a^2+bc&ab+bd\\ ca+dc&bc+d^2\end{pmatrix} \\=16\begin{pmatrix}b^2c^2+3a^2bc+bcd^2+2abcd+a^4&2ab^2c+2b^2cd+a^3b+bd^3+abd^2+a^2bd\\ 2abc^2+2bc^2d+a^3c+cd^3+acd^2+a^2cd&b^2c^2+a^2bc+3bcd^2+2abcd+d^4\end{pmatrix}$