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# Determinants

## Finding the Determinant

The determinant of an n x n matrix can be computed with a cofactor expression across any row or down any column.  

The cofactor expression across the ith row is given by this formula.

$$
det(A)=a\_{i1}C\_{i1}+a\_{i2}C\_{i2}\ ...\ a\_{in}C\_{in}
$$

where

$$
C\_{ij}=(-1)^{i+j}det(A\_{ij})
$$

where A\_ij is the **submatrix created by deleting the ith row and jth column of A.** As such, in the following example if A\_ij = 1, the following change would occur.

$$
Given A\_{ij}=1\\
\begin{bmatrix}
\colorbox{red}{1} & \colorbox{red}2 & \colorbox{red}3\\
\colorbox{red}4&5&6\\
\colorbox{red}7&8&9
\end{bmatrix}
\Longrightarrow
\begin{bmatrix}
5&6\\
8&9
\end{bmatrix} = A\_{ij}
$$

As for columns, it is given by this formula.

$$
det(A)=a\_{1j}C\_{2j}+a\_{2j}C\_{2j}\ ...\ a\_{nj}C\_{nj}
$$

C is the same as defined above.

You should get the same answer whatever method you choose.

{% hint style="info" %}
For matrices larger than 3x3, you will need to compute the cofactor expression multiple times to find the det(A) term, as you will always end up with det(A\_ij) where A has dimensions n-1 x n-1.
{% endhint %}

## Finding the Determinant II: The Diagonal Method

The downward diagonal method is an alternative method of finding the determinant for 3x3 matrices.

![Credit: Pearsons](https://2947382171-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-M6XPH5Qvmbnhx_A2DDH%2F-MAt9Yuz6NdALn98oX_U%2F-MAuI2GUhCXQIWu_9ET0%2FScreen%20Shot%202020-06-28%20at%202.39.52%20AM.png?alt=media\&token=67aaff98-86af-4df9-9216-bb3730c85cbf)

Multiply the quantities along the six diagonals, and then you can subtract the top ones and add the bottom ones to find the determinant.

## Properties of Determinants

Given A is a square matrix, and B is the matrix after the operations were performed on it:

* If one row is scaled and then added to another row (changing only one row) det A = det B.
* If two rows are interchanged, then det B = det A \* -1
* If a row is multiplied by scalar k, then det B = k \* det A.

There are additional properties that do not involve row operations.

*  For all square matrices, the determinant of the transpose of A is equal to det A.
* For two square matrices A and B, det AB = det A \* det B
* This go logically, but given k is constant, det kA = k det A.
* $$det A^T=detA$$