Linear Algebra/Determinant

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The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these properties:

  • It is linear on the rows of the matrix.
\det \begin{bmatrix}  \ddots & \vdots & \ldots \\ \lambda a_1 + \mu b_1 & \cdots & \lambda a_n + \mu b_n \\   \cdots & \vdots & \ddots \end{bmatrix} = \lambda \det \begin{bmatrix}  \ddots & \vdots & \cdots \\ a_1 & \cdots & a_n \\   \cdots & \vdots & \ddots \end{bmatrix} + \mu \det \begin{bmatrix}  \ddots & \vdots & \cdots \\  b_1 & \cdots & b_n \\   \cdots & \vdots & \ddots \end{bmatrix}
  • If the matrix has two equal rows its determinant is zero.
  • The determinant of the identity matrix is 1.

It is possible to prove that  \det A = \det A^T , making the definition of the determinant on the rows equal to the one on the columns.

Properties[edit]

  • The determinant is zero if and only if the rows are linearly dependent.
  • Changing two rows changes the sign of the determinant:
\det \begin{bmatrix} \cdots \\ \mbox{row A} \\ \cdots \\ \mbox{row B} \\ \cdots \end{bmatrix} = - \det \begin{bmatrix}\cdots \\ \mbox{row B} \\ \cdots \\ \mbox{row A} \\ \cdots \end{bmatrix}


  • The determinant is a multiplicative map in the sense that
\det(AB) = \det(A)\det(B) \, for all n-by-n matrices A and B.

This is generalized by the Cauchy-Binet formula to products of non-square matrices.

  • It is easy to see that \det(rI_n) = r^n \, and thus
\det(rA) = \det(rI_n \cdot A) = r^n \det(A) \, for all n-by-n matrices A and all scalars r.
  • A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have
\det(A^{-1}) = \det(A)^{-1}. \,

Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.

A matrix and its transpose have the same determinant:

\det(A^\top) = \det(A). \,

The determinants of a complex matrix and of its conjugate transpose are conjugate:

\det(A^*) = \det(A)^*. \,

Theorems[edit]

Uniqueness[edit]

Existance[edit]

Using Laplace's formula for the determinant

Binet's theorem[edit]

\det(A B) = \det A \cdot \det B

Applications[edit]