Comments on Compute the determinant
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Compute the determinant
Challenge
A simple challenge: Given a two-dimensional matrix (an array of arrays) of real numbers, compute the determinant.
The determinant of a matrix is a mathematical construct used in many applications, such as solving polynomial equations, identifying the invertibility of the matrix, and finding the scaling factor under a matrix transformation. For more information about it, see this Wikipedia entry.
There are a couple of different ways to compute the determinant, and it is up to you how you implement it.
For instance, you may compute it using the Laplace expansion, a recursive algorithm which goes
- Pick a row or column.
- Start with a sum of zero.
- For each entry of the row/column:
- Create a new matrix with the row and column of the entry removed. This new matrix is a square matrix of size one less that the original.
- Compute the determinant of that smaller matrix.
- Multiply that determinant by the entry.
- If the row index plus the column index is even[1], add it to the sum, otherwise, subtract it.
- The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
function laplaceDet(matrix) {
if (matrix.length === 1) return matrix[0][0];
let sum = 0;
for (let rowIndex = 0; rowIndex < matrix.length; ++rowIndex) {
let minorMatrix = matrix.filter((_, index) => index !== rowIndex)
.map(row => row.slice(1));
sum += ((-1) ** rowIndex) * matrix[rowIndex][0] * laplaceDet(minorMatrix);
}
return sum;
}
This is code-golf, so the program with the lowest byte-count wins!
Test cases
$$ \begin{aligned} \det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\ \det\begin{bmatrix}1.5&2\\-3&4.5\end{bmatrix}&=12.75 \\ \det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\ \det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\ \det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0 \end{aligned} $$
In text form,
[[1,0],[0,1]] -> 1
[[1.5,2],[-3,4.5]] -> 12.75
[[3,7],[1,-4]] -> -19
[[1,0,0],[0,1,0],[0,0,1]] -> 1
[[1,2,3],[4,5,6],[7,8,9]] -> 0
-
Note that it doesn't matter if it is 1-indexed or 0-indexed, as 1+1 and 0+0 are both the same parity. ↩︎
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Post
Python 3, 29 bytes
import numpy
numpy.linalg.det
Yeah, this is really boring, but I'm curious if this can be beat in Python.
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