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Determinant

Determinant is defined for square matrices and represents a transformation from \(\real^{n \times n} \to \real\). Its defined as follows for \(2 \times 2\) and \(3 \times 3\) matrices

\[ \begin{aligned} A &= [a]\newline \detm{A} &= a\newline A &= \begin{bmatrix} a &b\newline c &d \end{bmatrix}\newline \detm{A} &= ac - bd\newline A &= \begin{bmatrix} a &b &c\newline d &e &f\newline g &h &i \end{bmatrix}\newline \detm{A} &= a(ei-hf) - b(di - gf) + c(dh - ge) \end{aligned} \]

More precisely, read below on how the formula is derived

Minors

For any element \(a_{ij}\) of the matrix \(A\), the minor \(M_{ij}\) is the determinant of the matrix obtained after removing the \(i\) row and \(j\) column.

Since the determinant of a single element is the element itself, the minor of any element of a \(2 \times 2\) matrix is the diagonally opposite element.

Cofactors

The minor \(M_{ij}\) multiplied by \((-1)^{i+j}\) is called the cofactor \(C_{ij}\).

\[ \begin{aligned} C_{ij} = (-1)^{i+j}M_{ij} \end{aligned} \]

Adjoint

The transpose of the matrix obtained after replacing all the elements of the matrix with their cofactors is called adjoint, Adj(A). The adjoint is thus another matrix.

\[ \begin{aligned} Adj(A)\_{ij} = [C_{ij}]^{T} \end{aligned} \]
  • A \(\times\) Adj(A) = \(\detm{A} \times\) I
  • \(A^{-1} = \frac{1}{\detm{A}} Adj(A)\)

Calculating the Determinant

Let \(A\) be a \(n \times n\) matrix. Consider any row (or column) \(i\) of the matrix.

\[ \begin{aligned} \detm{A} = \sum_{j=1}^{n} a_{ij} C_{ij} \end{aligned} \]

Thus, the determinant can be obtained by expansion along any row or column. In calculation of the cofactors, the \(\pm\) sign will keep alternating as is evident from the above formula. We can verify that the definition for determinants of \(2 \times 2\) and \(3 \times 3\) can be derived from this formula.

Properties of Determinant

  • Changing rows and columns does not change the value of the determinant \(\detm{A^{T}} = \detm{A}\)
  • If any row or column of the matrix is zero, then \(\detm{A} = 0\)
  • If any two rows or columns of the matrix are interchanged, the determinant is multiplied by -1
  • If any two rows or columns of the matrix are identical, then \(\detm{A} = 0\) as well
    • follows from the last point
  • If all the elements of one row (or column) are multiplied by the same number k, the determinant is also multiplied by k
  • If \(A\) is \(n \times n\), then \(\detm{kA} = k^{n}\detm{A}\)
  • The sum of the products of the elements of any row (or column) with the cofactors of some other row (or column) is 0
    • sum of products with cofactors of the same elements is the determinant
  • The value of determinant is unchanged by addition of a scalar multiple of a row (or column) to another row (or column)
  • \(\detm{AB} = \detm{A} \detm{B}\)
    • It follows that \(\detm{A^{n}} = \detm{A}^{n}\)
    • \(\detm{A} \detm{A^{-1}} = 1\)
  • Determinant of an upper triangular or lower triangular or diagonal matrix is the multiplication of all the diagonal elements
  • Determinant of a skew-symmetric matrix of odd order is 0
  • \(\detm{Adj(A)} = \detm{A}^{n-1}\)
  • If the determinant of a matrix is non zero, then the matrix has full rank, or all its rows and columns are linearly independent, and the matrix is invertible.