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The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo (the latter determinant ...
There is a determinant map from the matrix ring GL(R ) to the abelianised unit group R × ab with the following properties: [1] The determinant is invariant under elementary row operations; The determinant of the identity matrix is 1; If a row is left multiplied by a in R × then the determinant is left multiplied by a
Determinant definition has only multiplication, addition and subtraction operations. Obviously the determinant is integer if all matrix entries are integer. However actual computation of the determinant using the definition or Leibniz formula is impractical, as it requires O(n!) operations.
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:
Let h be the unique increasing bijection [m] → S, and π,σ the permutations of [m] such that = and =; then () [], is the permutation matrix for π, (), [] is the permutation matrix for σ, and L f R g is the permutation matrix for , and since the determinant of a permutation matrix equals the signature of the permutation, the identity ...
The identity is now obtained by computing (′) in two ways. First, we can directly compute the matrix product ′ (using simple properties of the adjugate matrix, or alternatively using the formula for the expansion of a matrix determinant in terms of a row or a column) to arrive at
For example, when or is zero, we can eliminate the associated rows of the coefficient matrix without any changes to the rest of the output vector. If v {\displaystyle v} is null then the above equation for x {\displaystyle x} reduces to x = ( A − 1 + A − 1 B S − 1 C A − 1 ) u {\displaystyle x=\left(A^{-1}+A^{-1}BS^{-1}CA^{-1}\right)u ...