The determinant of a square matrix is a function mapping square matrices into real numbers, and it is an important theoretical tool in linear algebra. Actually, it was investigated before the introduction of the matrix concept. In Section 3.2.3 we have seen that determinants can be used to solve systems of linear equations by Cramer’s rule. Another use of the determinant is to check whether a matrix is invertible. However, the use of determinants quickly becomes cumbersome as their calculation involves a number of operations that increases very rapidly with the dimension of the matrix. Hence, the determinant is mainly a conceptual tool.
We may define the determinant inductively as follows, starting from a two-dimensional square matrix:
We already know that for a three-dimensional matrix, we may define the determinant by selecting a row and multiplying its elements by the determinant of the submatrix obtained by eliminating the row and the column containing that element:
The idea can be generalized to an arbitrary square matrix 
as follows:
- Denote by Aij the matrix obtained by deleting row i and column j of A.
- The determinant of this matrix, denoted by Mij ≡ det(Aij), is called the (i, j)th minor of matrix A.
- We also define the (i, j)th cofactor of matrix A as Cij ≡ (−l)i+jMij. We immediately see that a cofactor is just a minor with a sign. The sign is positive if i + j is even; it is negative if i + j is odd.
Armed with these definitions, we have two ways to compute a determinant:
- Expansion by row. Pick any row k, k = 1, …, n, and compute
- Expansion by column. Pick any column k, k = 1, …, n, and compute
The process can be executed recursively, and it results ultimately in the calculation of many 2 × 2 determinants. Indeed, for a large matrix, the calculation is quite tedious. The reader is invited to check that Eq. (3.11) is just an expansion along row 1.
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