Machinery's Handbook, 31st Edition
Matrix Operations 125 Transpose of a Matrix.— If the rows of a matrix A mn are interchanged with its columns, the new matrix is called the transpose of matrix A , or A T . The first row of the matrix be - comes the first column in the transposed matrix, the second row of the matrix becomes the second column, and the third row of the matrix becomes the third column. Example 3: A = A T = 1 4 7 2 5 8 3 6 9 Determinant of a Square Matrix.— Every square matrix A is associated with a real number, its determinant, which may be written det A or A . For A = a b c d , det A A = = ad – bc. 1 2 3 4 5 6 7 8 9
Example 4:
2 –1 1 –3
2 –1 1 –3
2 ( ) –3 ( )
1 ( ) –1 ( ) –
=
det A
=
=
=
–5
A
The process for taking the determinant of a 3 × 3 matrix is shown next. It entails multi- plying the first entry of each column by the determinant of the remaining 2 × 2 matrix and alternately adding or subtracting the product. a b c a b c
e f
d f
d e
d e f
d e f
B =
det B =
= a
– b
+ c
,
h i
g i
g h
g h i
g h i
Example 5: Find the determinant of the following matrix.
5 6 7 1 2 3 4 5 6
=
A
Solution:
det A 5 12– 15 ( ) 6 6– 12 ( ) – 7 5 – 8 ( ) + =
5 –3 ( ) 6 –6 ( ) – 7 –3 ( ) + = = –15+36–21 = 0 Minors and Cofactors.— The minor M ij of a matrix A is the determinant of a submatrix resulting from the elimination of row i and column j . If A is a square matrix, the minor M ij of the entry a ij is the determinant of the matrix obtained by deleting the i th row and j th column of A . The cofactor C ij of the entry a ij is given by C ij = ( - 1) ( i + j ) M ij . Thus, the sign of cofactor a ij alternates across the row it lies in. The matrix formed by its cofactors is called the cofactor matrix . Example 6: Find the minors and cofactors of A = Solution: To determine the minor M 11 , delete the first row and first column of A and find the determinant of the resulting matrix. 1 2 3 4 5 6 3 2 1
5 6 2 1
5 1 ( × )
6 2 ( × ) –
=
=
=
5– 12 –7 =
M 11
Similarly to find M 12 , delete the first row and second column of A and find the determi nant of the resulting matrix.
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