Question #282043
16[abcd]abcd(abcd)abcd16\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}


1
Expert's answer
2021-12-23T06:09:13-0500

Solution:

16(abcd)(abcd)(abcd)(abcd)=16(a2+bcab+bdca+dcbc+d2)(a2+bcab+bdca+dcbc+d2)=16(b2c2+3a2bc+bcd2+2abcd+a42ab2c+2b2cd+a3b+bd3+abd2+a2bd2abc2+2bc2d+a3c+cd3+acd2+a2cdb2c2+a2bc+3bcd2+2abcd+d4)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}


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