To find the determinant of a 2 by 2 matrix, first multiply the diagonal elements and the off-diagonal elements together. The diagonal elements of a 2 by 2 matrix are in the upper left-hand corner and the lower right-hand corner. Then, subtract the product of the off-diagonal elements from the product of the diagonal elements. This quantity is the determinant.
The determinant of a matrix is one method to find out if the matrix is invertible. If the determinant of a matrix A is not equal to zero, then A is invertible; likewise, if the determinant is equal to zero, then A is not invertible.
Calculating the determinant of a 3 by 3 matrix is more difficult than calculating for a 2 by 2 matrix, but the calculation can be done by hand. For larger matrices, the determinant is calculated using the property that the determinant of a diagonal matrix is equal to or the product of the elements on the diagonal. Using row and column permutations, transform the original matrix into a triangular matrix, and calculate the triangular determinant. The determinant of the original matrix is this triangular determinant multiplied by -1 for each time two rows were interchanged and by any constants that multiplied rows during the transformation.