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Appendix

In this appendix we first review the determinant of a matrix, then we define the adjoint

matrix, the inverse of a matrix, and the derivative and integral of a matrix.

Determinant of a Matrix. For each square matrix, there exists a determinant. The

determinant of a square matrix Ais usually written as or det A. The determinant has

the following properties:

1.If any two consecutive rows or columns are interchanged, the determinant changes

its sign.

2.If any row or any column consists only of zeros, then the value of the dererminant

is zero.

3.If the elements of any row (or any column) are exactly ktimes those of another

row (or another column), then the value of the determinant is zero.

4.If, to any row (or any column), any constant times another row (or column) is

added, the value of the determinant remains unchanged.

5.If a determinant is multiplied by a constant, then only one row (or one column) is

multiplied by that constant. Note, however, that the determinant of ktimes an

n*nmatrixAiskntimes the determinant of A, or

@kA@ =kn@A@

@A@

Vector-Matrix Algebra

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