Unformatted text preview: equation reduces to the definition in (8).
Expansion on the jth column (j=1,2,3).
det( ) = (−1)
+ (−1)
+ (−1) . . 11. Determinant of the matrix given in (8) by expansion along the second
column. ( j=3 )
4
det( ) = −2
7 6
1
+5
9
7 3
1
−8
9
4 3
= −2(−6) + 5(−12) − 8(−6) = 0
6 12. At this point, an inquisitive reader would raise the question why the
Laplace expansions at different rows and columns (totally 6 different ways)
yield the same value. For 3x3 matrix this can in fact be checked after
finitely many steps and calculations.
By expanding along the first row, a 3x3 determinant in general is equal to
− + which can be further expanded to
+ −
+
3 −
− , We can verify that the Laplace expansion at other row and other column
gives the same as the expression in the box above. From now on, we will
trust the fact in (10). There are six terms. Half of them have coefficient 1 and half of them
have coefficient 1.
Each term contain exactly one entry from each row and exactly one
entry from each column. The factors in each term in the box are
arranged so that the first subscripts are 1, 2 and 3. Written in this way,
the second subscripts in each term form a permutation of {1, 2, 3}.
Each of the 3!=6 permutations of {1, 2, 3} occurs exactly once, , , , , , . 13. The following properties of the determinant of a 3x3 matrix A = [aij] are
fundamental. (A) The determinant of the identity matrix is 1. (B1) With columns 2 and 3 held fixed, the determinant function is linear as
a function of the first column, meaning that
det ∙
∙
∙ +
+
+ ∙
∙
∙
= ∙ det + ∙ det . (B2) With columns 1 and 3 fixed, the determinant is linear as a function of
the second column.
(B3) With columns 1 and 2 fixed, the determinant is linear function of the
third column. 4 (C) If two columns are identical, the determinant is zero. For example, if the
second and third column of A are the same, we expand on the first column
and get
det = − + Each of the three 2x2 determinants is zero, and hence the LHS is also zero. (D) If we interchange two columns in A, the determinant is multiplied by 1.
As in (C), we can also see this by “reduction” to the 2x2 case. For example,
we want to exchange the first and second column of A. We expand along
the third column,
= − =− + + = − − . In the second equality, we have used the fact that after exchanging the two
columns in a 2x2 determinant, we change the sign of the determinant. The
first and last equalities follow from Laplace expansion. (E) det(A) = det(AT). This can be proved by directly applying Laplace
expansion to A and AT, and then comparing the results.
Since A and AT have the same determinant, properties (B), (C) and (D)
holds with “column” replaced by “row ”. 14. Adjoint. Let be the matrix
A
= −A
A −A
A
−A A
−A
A where Aij is the minor obtained by removing row i and column j. Define the
adjoint of A as the transpose of
adj( ) = ,
A
= −A
A −A
A
−A A compact expression for adj(A) is [(1)i...
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 Spring '13
 Invertible matrix, Det

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