If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Intro to determinant notation and computation

Matrix determinants are easy to define and hard to understand. So let's start with defining them and introducing related notation. In other videos we will learn what they mean and how to use them. Created by Sal Khan.

## Want to join the conversation?

• Can you only find matrix determinants in 2x2 matrices?

If not, then what method would you use to find the matrix determinant in matrices with different dimensions?
• It is actually possible to find determinants in other matrices (not 2 by 2). For example, if you wanted to find the determinant of a 1 by 1 matrix, the answer would just be that same number. However, if you were to find a matrix in the form of 2 by 3, 3 by 5, etc..., it would be impossible to find the determinant. As long as long as you are looking at a matrix in the form of x by x, where both values of x are equivalent, it is possible to find the determinant. However, from matrix to matrix in that form, finding the determinant varies.
• But what is the determinant?
• A determinant is essentially the scalar quantity which is found in a matrix obtaining the form of x by x, where both values of x are the same (square matrix).
• How do you calculate the determinant of a matrix bigger than 2x2?
(1 vote)
• To compute the determinant of a matrix larger than 2x2, you can use a process called expansion by minors or cofactor expansion. Here are the general steps:

Choose any row or column of the matrix.
For each element in that row or column, compute its corresponding minor, which is the determinant of the submatrix obtained by deleting the row and column that the element is in.
For each element in that row or column, multiply it by its corresponding minor and its sign, which is positive or negative depending on its position in the matrix. Specifically, the sign of an element in row i and column j is (-1)^(i+j).
Sum up all the products obtained in step 3 to get the determinant of the original matrix.
This process may look daunting for larger matrices, but it can be simplified by choosing a row or column that has many zeros or that has a repeated pattern. Additionally, there are some properties of determinants, such as linearity and multiplicativity, that can make the computation easier in some cases.