3A.6 Determinants

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Now, let us address several analytical issues of the determinant of a square matrix. Consider the matrix

A ¼

a11 · · · a1n

.. .

an1 · · · ann

2

6664

3

7775

The minor of aij ¼ Mij ¼ the determinant of matrix formed by deleting the ith row and the jth column of

the original matrix.

The cofactor of aij ¼ Cij ¼ ð21ÞiþjMij

cof ðAÞ ¼ cofactor matrix of A

adjðAÞ ¼ adjoint A ¼ ðcof AÞT

3A.6.1 Properties of Determinant of a Matrix

1. Interchange two rows (columns) ) determinant’s sign changes.

2. Multiply one row (column) by a ) a det( ).

3. Add a [a £ row(column)] to a second row(column) ) determinant unchanged.

4. Identical rows(columns) ) zero determinant.

5. For two square matrices A and B; detðABÞ ¼ detðAÞdetðBÞ:

3A.6.2 Rank of a Martix

Rank A ¼ number of linearly independent columns ¼ number of linearly independent rows ¼

dim(column space) ¼ dim(row space)

Here “dim” denotes the “dimension of.”

Modal Analysis 3-53

© 2005 by Taylor & Francis Group, LLC

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