Definition
With each square matrix corresponds just one number. This
number is called the determinant of the matrix.
The
determinant of a matrix A is denoted det(A) or |A|.
Determinant of a 1 x 1 matrix
Determinant
of the matrix is the element itself.
Ex:
det([-7]) = -7
Determinant of
a 2x2 matrix and 3x3 matrix
a 2x2 matrix and 3x3 matrix
Do not memorize these as formulas, learn instead the
patterns which give the terms. The 2 x 2 case is easy: the product of the
elements on one diagonal (the "main diagonal"), minus the product of
the elements on the other (the "antidiagonal").
For the 3 x 3 case, three products get the +
sign: those formed from the main diagonal,
or having two factors parallel to the main diagonal. The
other three get a negative sign: those from the antidiagonal, or having two factors
parallel to it.
Important facts about
A
ü A is
multiplied by -1 if we interchange two rows or two columns.
ü A = 0
if one row or column is all zero, or if two rows or two columns are the same.
ü A is multiplied
by c if every element of some row or column is multiplied by c.
ü The value of
A is unchanged if we add to one
row (or column) a constant multiple of another row (resp. column).
Important facts about
A
Though the letters a, b, c,.. . can be used for very small
determinants, they can't for larger ones; it's important early on to get used
to the standard notation for the entries of determinants. This is what the
common software packages and the literature use. The determinants of order two
and three would be written respectively
In general, the ij-entry, written aij, is the
number in the i-th row and j-th column.
Its ij-minor, written I Aij I , is
the determinant that's left after deleting from IAl the row and column
containing aij.
Its ij-cofactor, written here Aij
,is given as a formula by Aij = (-1)i+j lAijl.
Laplace expansion by cofactors
Select any
row (or column) of the determinant. Multiply each entry aij in that
row (or column) by its cofactor Aij, and add the three resulting
numbers; you get the value of the determinant.
As practice
with notation, here is the formula for the Laplace expansion of a third order
(i.e., a 3 x 3) determinant using the cofactors of the first row:
and the
formula using the cofactors of the j-th column:
Example..
Evaluate the determinant of matrix:
by using Laplace expansion!
The laplace expansion by the first row is
The laplace expansion by the second column is:
Example..
Area & Volume Interpretation of Determinant
Cramer’s Rule à The use of determinant
to solve the linear equations system
TUGAS IV
- Cari solusi persamaan linier berikut dengan cramer’s rule!
2 x - 3 y - z = -1
x +
y - z = 2
4 x - 3 y -
2 z = -1
2. Dengan a,b,c,d,e,f,g,h adalah angka di stambukmu
(Dabcdefgh1),
hitung determinan dari matriks berikut dengan
metode laplace:
a b c
d e f
g h 1
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