The determinant of a 3x3 matrix A, denoted as det(A) or |A|, is a scalar value that can be calculated from the elements of the matrix.
The formula for the determinant of a 3x3 matrix is:
| a b c |
| d e f | = a*(ei - fh) - b*(di - fg) + c*(dh - eg)
| g h i |
where a, b, c, d, e, f, g, h, i are the elements of the matrix A.
It can also be computed using the rule of Sarrus which is a mnemonic device to remember the formula for 3x3 determinant:
| a b c |
| d e f | = aei + bfg + cdh - afh - bdi - ceg
| g h i |
The determinant of a matrix has many uses in linear algebra and it is used to determine the invertibility of a matrix, it is also used to calculate the area or volume of a parallelepiped spanned by the columns or rows of the matrix.
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