Inverse of a Matrix

Inverse of a Matrix

 

For a given square matrix A = ǀǀaijǀǀn1 of order n there exists a matrix B = ǀǀbijǀǀn1 of the same order (called inverse matrix) such that AB = E, where E is the unit matrix; then the equation BA = E also holds. The inverse of a matrix A is designated as A–1. For the existence of the inverse of a matrix A–1, it is necessary and sufficient that the determinant of the given matrix A be nonzero; that is, the matrix A must be nonsingular. The elements bij of the inverse of a matrix are found by the formula bij = Aji/D, where Aji is the cofactor of the element aij of matrix A and D is the determinant of matrix A.

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Where the matrix O = AT-AT BB+ and (OA) + represent the Moore-Penrose generalized inverse of a matrix (OA).
Secondary school teachers may use this activity to consolidate their students' learning of certain concepts of matrices such as the algorithm for matrix multiplication and the concept of the multiplicative inverse of a matrix.
For instance, teachers can teach students to make use of the functions such as MMULT, MDETER and MINVERSE in Microsoft Excel to perform matrix multiplication, as well as to find the determinant and inverse of a matrix.
Subroutines for Testing Programs that Computer the Generalized Inverse of a Matrix.
The case q = -1, which leads to estimates of the trace of the inverse of a matrix, was studied in [4]; on this problem, see [10].
This idea was introduced in [3] for estimating the norm of the error in the solution of a system of linear algebraic equations, and it was used in [4] for estimating the trace of the inverse of a matrix.
1](z) given in [4], which leads to the estimate of the trace of the inverse of a matrix.
which leads to the two-term estimate given in [4] for the trace of the inverse of a matrix.
4, the structure of the inverse of a matrix F is given by the transitive closure of the directed graph of the transposed matrix [F.
4 that the directed graph of the inverse of a matrix is the transitive closure of the directed graph corresponding to the matrix.
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