Matrices : Determinant

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The determinant command in the matrices section of QuickMath allows you to find the determinant of any square matrix (a matrix which has the same number of rows and columns).

The determinant of a square matrix A can be calculated in terms of its elements ai,j and the determinants of the smaller matrices Ai,j.

For example, if A is a 3 x 3 matrix, then its determinant can be found as follows :

det(A) = a1,1 |A1,1| - a1,2 |A1,2| + a1,3 |A1,3|

where ai,j is the element of A at row i, column j and Ai,j is the matrix constructed from A by removing row i and column j.

The determinant of a 2 x 2 matrix

a b
c d

can be easily memorized, rather than calculated from scratch. It is

a d - b c.

To use the determinant command, simply go to the determinant page, type in your matrix and hit the "Determinant " button. Your question will be automatically answered by computer and the reply will be shown in your browser within a few seconds. The answer will be given in two forms : a 'natural' form, which mimics the row and column layout normally used for matrices, and an 'input' form, which is the form required for the entry of matrices into QuickMath. The input form of a solution can be copied and pasted into a matrix input field, so you can use the result from one calculation in another calculation without having to re-type it.


Here are some examples illustrating the types of matrices you can use the determinant command on and the results which QuickMath will return.

Matrix Result
a, b
c, d
-(b c) + a d
1, 2
3, 4
5, 7, 3, 4
8, 9, 2, 3
1, 5, 3, 8
3, 6, 9, 0