Determinant of 4×4 Matrix
Determinant of a 4×4 matrix is a unique number which is calculated using a particular formula. If a matrix order is n x n, then it is a square matrix. Hence, here 4×4 is a square matrix which has four rows and four columns. If A is square matrix then the determinant of matrix A is represented as |A|.
To find the determinant of a 4×4 matrix, we will use the simple method, which we usually use to find the determinant of a 3×3 matrix.
How to calculate determinant of 4×4 matrix?
Before we calculate the determinant of a matrix of order 4, let us first check a few conditions.
- if there is any condition, where determinant could be 0 (for example, the complete row or complete column is 0)
- if factoring out of any row or column is possible.
- If the elements of the matrix are the same but reordered on any column or row.
If any of the three cases given above is met, the corresponding methods for calculating 3×3 determinants are used. We transform a row or a column to fill it with 0, except for one element. The determinant will be equivalent to the product of that element and its cofactor. In this situation, the cofactor is a 3×3 determinant, which is estimated with its particular formula.
Let us solve some examples here.
As we can see in the above example, the elements in third row is all 0. Hence, the value of determinant will be zero.
As we can see here, column C1 and C3 are equal. Therefore, the determinant of the matrix is 0.
As we can see here, second and third rows are proportional to each other. Hence, the determinant of the matrix is 0.
As we can see, there is only one element other than 0 on first column, therefore we will use the general formula using this column. The cofactors of the elements which are 0 are not required to be evaluated because the product of cofactors and the elements will be 0 here.