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Sandbox Compute the determinant

posted 3y ago by Moshi‭ · edited 3y ago by Moshi‭

Article code-golf math

#4: Post edited by

Moshi‭ · 2021-08-22T19:33:04Z (over 3 years ago)

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## 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](https://en.wikipedia.org/wiki/Determinant).
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](https://en.wikipedia.org/wiki/Laplace_expansion), a recursive algorithm which goes
1. Pick a row or column.
2. Start with a sum of zero.
3. For each entry of the row/column:
1. 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.
2. Compute the determinant of that smaller matrix.
3. Multiply that determinant by the entry.
4. If the row index plus the column index is even[^1], add it to the sum, otherwise, subtract it.
4. The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##VVFRT4MwEH5uf8X5ZBmFgNucBtmTL5qoDz4SYggrswbaBTqdMfvt81rG2BpSet939/W761fxXXRlKzcmUHolDodqq0ojtYK62NRFKR6FYU1hWrnz4I8SWcExDGuh1uYT0jSF2INWmG2roOeyKMcvoZTUwkC3bSCFKKGk0i0wC7X650mtxM7hY/QAF9oJ@P7AududXCOVbl9cHpYfCypZG9Ey9sFB9unpsj/BFRo8qaAGgdMKm2LDkLPJ@Au7WpaCxZ6HXom17afAWIDtTSajBkyGNgfIdovo@chGk1ZsT8lxQKia0P2h1KozYERnOmwioyTLYg5RzimBLOIQ53h0aDjncNPjwZTDLJwPFEaLnsDSYHaqQJ0LqfPgQhqFOUx7bsYBL7rtgwWHOw73eU5zSt2jjX5BV71v9yIW17UIa71mz@9vr2GHLau1rH6ZTfLAh2sIlrj55@NxnGcH8Q8 "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!
## Test cases
$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
~~\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\~~
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$
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
```
## Questions
- Anything else I missed?
[^1]: 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.

## 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](https://en.wikipedia.org/wiki/Determinant).
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](https://en.wikipedia.org/wiki/Laplace_expansion), a recursive algorithm which goes
1. Pick a row or column.
2. Start with a sum of zero.
3. For each entry of the row/column:
1. 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.
2. Compute the determinant of that smaller matrix.
3. Multiply that determinant by the entry.
4. If the row index plus the column index is even[^1], add it to the sum, otherwise, subtract it.
4. The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##VVFRT4MwEH5uf8X5ZBmFgNucBtmTL5qoDz4SYggrswbaBTqdMfvt81rG2BpSet939/W761fxXXRlKzcmUHolDodqq0ojtYK62NRFKR6FYU1hWrnz4I8SWcExDGuh1uYT0jSF2INWmG2roOeyKMcvoZTUwkC3bSCFKKGk0i0wC7X650mtxM7hY/QAF9oJ@P7AududXCOVbl9cHpYfCypZG9Ey9sFB9unpsj/BFRo8qaAGgdMKm2LDkLPJ@Au7WpaCxZ6HXom17afAWIDtTSajBkyGNgfIdovo@chGk1ZsT8lxQKia0P2h1KozYERnOmwioyTLYg5RzimBLOIQ53h0aDjncNPjwZTDLJwPFEaLnsDSYHaqQJ0LqfPgQhqFOUx7bsYBL7rtgwWHOw73eU5zSt2jjX5BV71v9yIW17UIa71mz@9vr2GHLau1rH6ZTfLAh2sIlrj55@NxnGcH8Q8 "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, 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
```
## Questions
- Anything else I missed?
[^1]: 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.

#3: Post edited by

Moshi‭ · 2021-08-22T19:23:23Z (over 3 years ago)

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## Challenge
~~A simple challenge: Given a two-dimensional matrix (an array of arrays) of integers, 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](https://en.wikipedia.org/wiki/Determinant).
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](https://en.wikipedia.org/wiki/Laplace_expansion), a recursive algorithm which goes
1. Pick a row or column.
2. Start with a sum of zero.
3. For each entry of the row/column:
1. 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.
2. Compute the determinant of that smaller matrix.
3. Multiply that determinant by the entry.
4. If the row index plus the column index is even[^1], add it to the sum, otherwise, subtract it.
4. The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##XVHRToMwFH2mX3F9soxCwE2nQfbkiybqg4@EGMLKrCntUjqdMfv22VIYTNLQe8@5PT339rP8KttKsa0OhVzT47HeiUozKYCXW15W9IFq3JRasb0Pv8hjNfRpxKnY6A/IsgwSHxTVOyXAcXlcmJUi5HGqod01kEGcIq@WCrCFlPx@FGu67/Axu4cz7RSCYOC62zu5hgmpnrs6c7w/UDOuqcL4nQBz5dnKRXBhDJ5UjIYHpy9qyi02nC02W9RyVlGc@L7x6lnbQQYYh6a92WzUgNnQ5gDZbg06Hdlo0oodkNcPyKim6HCspGg1aNrq1jSRIy/PEwJxQZAHeUwgKUzYo1cOnRNYDKiJlw41fLiYFMdnKtPkvyqBueMWBK4J3LhkSeCWwF1RoAKh7r1GqyBrZ7l7DItLTiMuN/jp7fUlak23YsPqH2yLfAjgEsKV@QXTyXScb2fwBw "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!
## Test cases
$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$
In text form,
```
[[1,0],[0,1]] -> 1
~~[[1,2],[3,4]] -> -2~~
[[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
```
## Questions
~~- Are there any alternative inputs that should be allowed?~~
~~- Should I expand it to all real numbers, not just integers?~~
- Anything else I missed?
[^1]: 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.

## 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](https://en.wikipedia.org/wiki/Determinant).
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](https://en.wikipedia.org/wiki/Laplace_expansion), a recursive algorithm which goes
1. Pick a row or column.
2. Start with a sum of zero.
3. For each entry of the row/column:
1. 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.
2. Compute the determinant of that smaller matrix.
3. Multiply that determinant by the entry.
4. If the row index plus the column index is even[^1], add it to the sum, otherwise, subtract it.
4. The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##VVFRT4MwEH5uf8X5ZBmFgNucBtmTL5qoDz4SYggrswbaBTqdMfvt81rG2BpSet939/W761fxXXRlKzcmUHolDodqq0ojtYK62NRFKR6FYU1hWrnz4I8SWcExDGuh1uYT0jSF2INWmG2roOeyKMcvoZTUwkC3bSCFKKGk0i0wC7X650mtxM7hY/QAF9oJ@P7AududXCOVbl9cHpYfCypZG9Ey9sFB9unpsj/BFRo8qaAGgdMKm2LDkLPJ@Au7WpaCxZ6HXom17afAWIDtTSajBkyGNgfIdovo@chGk1ZsT8lxQKia0P2h1KozYERnOmwioyTLYg5RzimBLOIQ53h0aDjncNPjwZTDLJwPFEaLnsDSYHaqQJ0LqfPgQhqFOUx7bsYBL7rtgwWHOw73eU5zSt2jjX5BV71v9yIW17UIa71mz@9vr2GHLau1rH6ZTfLAh2sIlrj55@NxnGcH8Q8 "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!
## Test cases
$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$
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
```
## Questions
- Anything else I missed?
[^1]: 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.

#2: Post edited by

Moshi‭ · 2021-08-19T22:40:16Z (over 3 years ago)

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## Challenge
A simple challenge: Given a two-dimensional matrix (an array of arrays) of integers, compute the determinant.
There are a couple of different ways to compute the determinant, and it is up to you how you implement it. As an example, here is an ungolfed implementation using the [Laplace expansion](https://en.wikipedia.org/wiki/Determinant#Laplace_expansion) along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##XVHRToMwFH2mX3F9soxCwE2nQfbkiybqg4@EGMLKrCntUjqdMfv22VIYTNLQe8@5PT339rP8KttKsa0OhVzT47HeiUozKYCXW15W9IFq3JRasb0Pv8hjNfRpxKnY6A/IsgwSHxTVOyXAcXlcmJUi5HGqod01kEGcIq@WCrCFlPx@FGu67/Axu4cz7RSCYOC62zu5hgmpnrs6c7w/UDOuqcL4nQBz5dnKRXBhDJ5UjIYHpy9qyi02nC02W9RyVlGc@L7x6lnbQQYYh6a92WzUgNnQ5gDZbg06Hdlo0oodkNcPyKim6HCspGg1aNrq1jSRIy/PEwJxQZAHeUwgKUzYo1cOnRNYDKiJlw41fLiYFMdnKtPkvyqBueMWBK4J3LhkSeCWwF1RoAKh7r1GqyBrZ7l7DItLTiMuN/jp7fUlak23YsPqH2yLfAjgEsKV@QXTyXScb2fwBw "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!
## Test cases
$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$
## Questions
- Are there any alternative inputs that should be allowed?
- Should I expand it to all real numbers, not just integers?
~~- Anything else I missed?~~

## Challenge
A simple challenge: Given a two-dimensional matrix (an array of arrays) of integers, 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](https://en.wikipedia.org/wiki/Determinant).
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](https://en.wikipedia.org/wiki/Laplace_expansion), a recursive algorithm which goes
1. Pick a row or column.
2. Start with a sum of zero.
3. For each entry of the row/column:
1. 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.
2. Compute the determinant of that smaller matrix.
3. Multiply that determinant by the entry.
4. If the row index plus the column index is even[^1], add it to the sum, otherwise, subtract it.
4. The final sum is the determinant.
As an example, here is an ungolfed implementation along the first column.
```javascript
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;
}
```
[Try it online!](https://tio.run/##XVHRToMwFH2mX3F9soxCwE2nQfbkiybqg4@EGMLKrCntUjqdMfv22VIYTNLQe8@5PT339rP8KttKsa0OhVzT47HeiUozKYCXW15W9IFq3JRasb0Pv8hjNfRpxKnY6A/IsgwSHxTVOyXAcXlcmJUi5HGqod01kEGcIq@WCrCFlPx@FGu67/Axu4cz7RSCYOC62zu5hgmpnrs6c7w/UDOuqcL4nQBz5dnKRXBhDJ5UjIYHpy9qyi02nC02W9RyVlGc@L7x6lnbQQYYh6a92WzUgNnQ5gDZbg06Hdlo0oodkNcPyKim6HCspGg1aNrq1jSRIy/PEwJxQZAHeUwgKUzYo1cOnRNYDKiJlw41fLiYFMdnKtPkvyqBueMWBK4J3LhkSeCWwF1RoAKh7r1GqyBrZ7l7DItLTiMuN/jp7fUlak23YsPqH2yLfAjgEsKV@QXTyXScb2fwBw "JavaScript (Node.js) – Try It Online")
This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!
## Test cases
$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$
In text form,
```
[[1,0],[0,1]] -> 1
[[1,2],[3,4]] -> -2
[[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
```
## Questions
- Are there any alternative inputs that should be allowed?
- Should I expand it to all real numbers, not just integers?
- Anything else I missed?
[^1]: 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.

#1: Initial revision by

Moshi‭ · 2021-08-15T02:51:00Z (over 3 years ago)

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Compute the determinant

## Challenge

A simple challenge: Given a two-dimensional matrix (an array of arrays) of integers, compute the determinant.

There are a couple of different ways to compute the determinant, and it is up to you how you implement it. As an example, here is an ungolfed implementation using the [Laplace expansion](https://en.wikipedia.org/wiki/Determinant#Laplace_expansion) along the first column.

```javascript
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;
}
```

[Try it online!](https://tio.run/##XVHRToMwFH2mX3F9soxCwE2nQfbkiybqg4@EGMLKrCntUjqdMfv22VIYTNLQe8@5PT339rP8KttKsa0OhVzT47HeiUozKYCXW15W9IFq3JRasb0Pv8hjNfRpxKnY6A/IsgwSHxTVOyXAcXlcmJUi5HGqod01kEGcIq@WCrCFlPx@FGu67/Axu4cz7RSCYOC62zu5hgmpnrs6c7w/UDOuqcL4nQBz5dnKRXBhDJ5UjIYHpy9qyi02nC02W9RyVlGc@L7x6lnbQQYYh6a92WzUgNnQ5gDZbg06Hdlo0oodkNcPyKim6HCspGg1aNrq1jSRIy/PEwJxQZAHeUwgKUzYo1cOnRNYDKiJlw41fLiYFMdnKtPkvyqBueMWBK4J3LhkSeCWwF1RoAKh7r1GqyBrZ7l7DItLTiMuN/jp7fUlak23YsPqH2yLfAjgEsKV@QXTyXScb2fwBw "JavaScript (Node.js) – Try It Online")


This is <a class="badge is-tag" href="/categories/49/tags/4274">code-golf</a>, so the program with the lowest byte-count wins!

## Test cases

<p>$$
\begin{aligned}
\det\begin{bmatrix}1&0\\0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2\\3&4\end{bmatrix}&=-2 \\
\det\begin{bmatrix}3&7\\1&-4\end{bmatrix}&=-19 \\
\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}&=1 \\
\det\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}&=0
\end{aligned}
$$</p>

## Questions

 - Are there any alternative inputs that should be allowed?
 - Should I expand it to all real numbers, not just integers?
 - Anything else I missed?

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