Machinery's Handbook, 31st Edition
126
Matrix Operations
4 6 3 1
4 1 ( × )
6 3 ( × ) –
=
=
=
4– 18 –14 =
M 12
Continuing this way, we obtain the following minors: M 11 = –7 M 12 = –14
M 13 = –7 M 23 = –4 M 33 = –3
M 21 = –4 M 31 = –3
M 22 = –8 M 32 = –6 ( i + j ) × M
To find the cofactor, calculate C ij = ( - 1) Similarly C 12 = ( - 1) (1+2) × M
ij , thus C 11 = ( - 1)
11 = 1 × ( - 7) = - 7.
(1+1) × M
12 = ( - 1)( - 14) = 14, and continuing this way we obtain the
following cofactors
C 11 = –7 C 21 = 4 C 31 = –3
C 12 = 14 C 22 = –8 C 32 = 6
C 13 = –7 C 23 = 4 C 33 = –3
–7 14 –7
Thus, the cofactor matrix is
4 –8 4
–3 6 –3 Adjoint of a Matrix.— The transpose of cofactor matrix is called the adjoint matrix. To obtain the adjoint matrix, the cofactor matrix is determined and then transposed. Example 7: Find the adjoint matrix of A :
1 2 3 4 5 6 3 2 1
=
A
Solution: The cofactor matrix from the above example is shown below at the left, and the adjoint matrix is shown on the right.
T
–7 14 –7 4 –8 4 –3 6 –3
–7 14 –7 4 –8 4 –3 6 –3
–7 4 –3 14 –8 6 –7 4 –3
cofactor A ( )
adj A ( )
=
=
=
Singularity and Rank of a Matrix.— A singular matrix is one whose determinant is zero. The rank of a matrix is the maximum number of linearly independent row or column vectors it contains. In ALGEBRA , Solving a System of Linear Equations , it is explained that a system with a unique solution is a linearly independent system. Such systems are vital to depicting many real-life processes. Dependent systems are those with infinite solutions (the lines are collinear), and their determinant is zero. Linearly independent vectors have a non-zero determinant. Inverse of a Matrix.— A square non-singular matrix A has an inverse A - 1 such that the product of matrix A and inverse matrix A - 1 is the identity matrix I . The operation is com- mutative as well. Thus, AA - 1 = A - 1 A = I . The inverse is the ratio of adjoint of the matrix and the determinant of that matrix. A –1 adj A ( ) A = -------- Example 8: What is the inverse of the following matrix?
2 3 5 4 1 6 1 4 0
=
A
Solution: The basic formula of an inverse of a matrix is A –1 adj A ( ) A = --------
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