4x4 Matrix
Find the determinant of the shown 4 x 4 matrix:
Expand Hint
$$$\begin{bmatrix}a & b & c & d\\ e & f & g & h\\ i & j & k & l\\ m & n & o & p\end{bmatrix}=a\cdot \begin{bmatrix}f & g & h\\ j & k & l\\ n & o &p \end{bmatrix}-b\cdot \begin{bmatrix}e & g & h\\ i & k & l\\ m & o & p\end{bmatrix}+c\cdot \begin{bmatrix}e & f & h\\ i & j & l\\ m & n & p\end{bmatrix}-d\cdot \begin{bmatrix}e & f & g\\ i & j & k\\ m & n & o\end{bmatrix}$$$
Hint 2
Remember, to solve a 3 x 3 matrix:
$$$\begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$$$
The formula to solve a 4 x 4 matrix, which breaks down the problem to multiple 3 x 3 matrices:
$$$\begin{bmatrix}a & b & c & d\\ e & f & g & h\\ i & j & k & l\\ m & n & o & p\end{bmatrix}=a\cdot \begin{bmatrix}f & g & h\\ j & k & l\\ n & o &p \end{bmatrix}-b\cdot \begin{bmatrix}e & g & h\\ i & k & l\\ m & o & p\end{bmatrix}+c\cdot \begin{bmatrix}e & f & h\\ i & j & l\\ m & n & p\end{bmatrix}-d\cdot \begin{bmatrix}e & f & g\\ i & j & k\\ m & n & o\end{bmatrix}$$$
Remember, to solve a 3 x 3 matrix:
$$$\begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$$$
For our problem statement:
$$$\begin{bmatrix}0 & 0 & 0 & 0\\ 12 & 11 & 10 & 9\\ 8 & 7 & 6 & 5\\ 4 & 3 & 2 & 1\end{bmatrix}=0\cdot \begin{bmatrix}11 & 10 & 9\\ 7 & 6 & 5\\ 3 & 2 & 1\end{bmatrix}-0\cdot \begin{bmatrix}12 & 10 & 9\\ 8 & 6 & 5\\ 4 & 2 & 1\end{bmatrix}+0\cdot \begin{bmatrix}12 & 11 & 9\\ 8 & 7 & 5\\ 4 & 3 & 1\end{bmatrix}-0\cdot \begin{bmatrix}12 & 11 & 10\\ 8 & 7 & 6\\ 4 & 3 & 2\end{bmatrix}$$$
Thus,
$$$\begin{bmatrix}0 & 0 & 0 & 0\\ 12 & 11 & 10 & 9\\ 8 & 7 & 6 & 5\\ 4 & 3 & 2 & 1\end{bmatrix}=0-0+0-0=0$$$
0
Time Analysis
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