Now I’m going to explain how to calculate the determinant of a square matrix using its elements.
Let’s see it!
How to calculate the determinant of a square matrix of order 3
The determinant of a square matrix is equal to the sum of the products of the elements of any row or any column, by their respective attachments.
For a matrix A of order 3:
Its determinant, choosing for example row 2 would be:
Let’s see it with an example. Let’s calculate the determinant of the following matrix:
The determinant of the matrix is:
We choose row 1 and add each element of that row by its attachment:
We operate the powers and solve the determinants of order 2 that we have:
And we operate:
To calculate the determinant of a matrix, you can choose any row or any column. To prove it, we are going to calculate the same matrix A, but now choosing row 3:
We make the sum of the products of each element of row 3 by its attachment:
We solve powers and determinants and operate:
The result is the same as when we chose row 1, as it could not be otherwise, but this time, we have had to perform fewer calculations, since being 0 one of its elements, this term is cancelled.
Therefore, it is convenient to choose the row or column that has more zeros.
Another option could have been to make zero all the elements except one of the row or the column that is chosen, by means of operations between their rows previously.
Let’s see how to do this, while I explain the calculation of the determinant of a square matrix of order 4.
Calculation of the determinant of a square matrix of order 4 (or higher)
The calculation of the determinant of square matrices of order 4 or higher is carried out following the same procedure, that is to say, a row or any column is chosen and the sum of the products of each element of the row or column is carried out by its attachment:
The determinant of matrix A above, choosing row 2 would be:
However, in this case, the calculation becomes more tedious, as we would have to solve in the worst case 4 determinants of order 3, and take all the steps we have taken in the previous section for each of those determinants.
When the determinants are of order 4 or higher, it is convenient to carry out internal operations with their rows, to make zeros all the elements except one, of the row or column we choose.
Let’s see it with an example: Calculate the determinant of the following matrix:
Whose determinant is:
Taking advantage that I have a 1 in the first element of row 1, I’m going to make 0 the rest of the elements of column 1, through internal operations. For it to the row 2, the rest 5 times the row 1:
Row 3 is left 3 times row 1:
And to row 4 I add row 1:
I am left with the next determinant:
Now, I am going to calculate the determinant, choosing column 1 since I will only have to multiply the first element by its adjunct to be the rest of elements 0:
I only have to solve a determinant of order 3. I can solve it as we have done in the previous section, but this time, I will solve it following the same procedure of making 0 the elements of a row or column.
Taking advantage that I have a -1 in the first element of row 2, I am going to make 0 the rest of the elements of column 1. To do this, the row 1 will rest 15 times the row 2:
And to row 3 I add 5 times row 2:
I am left with the determinant as follows:
To calculate this determinant, I choose column 1, so I will only have to multiply -1 by its attachment, which in turn multiplies everything I already had:
I solve the power and the determinant of order 2 that I have left:
And finally I operate until the solution is reached: