1. Main Use of Determinants
- They test for invertibility. If the determinants of A is zero, then A is singular. If detA ≠0, then A is invertible.
- The determinant of A equals the volume of a box in n-dimensional space. The edges of the box come from the rows of A. The columns of A would give an entirely different box with the same volume.
- The determinants gives a formula for each pivots.
- The determinant measures the dependence of
on each element of b. If one parameter is changed in an experiment, or one observation is corrected, the "influence coefficient" in
is a ratio of determinants.
2. Properties of the Determinant
- The determinant of the identity matrix is 1
- The determinant changes sign when two rows are exchanged.
The determinant of every permutation matrix is det P=±1. By row exchanges, we can turn P into the identity matrix. - The determinant is linear in each row separately
- If two rows of A are equal, then detA =0
- Subtracting a multiple of one row from another row leaves the same determinant. (The usual elimination steps do not affect the determinant)
- If A has a row of zeros, then det A = 0
- If A is triangular then det A is the product
of diagonal entries. If triangular A has 1s along the diagonal, then det A = 1
- If A is singular, then det A = 0. If A is invertible , then det A ≠ 0.
- The determinant of AB is the product of det A and det B
product rule: |A||B|=|AB| - The transpose of A has the same determinant as A itself:
From this point, every rule that applied to the rows can now be applied to the columns: The determinant change sign when two columns are exchanged, two equal columns (or a column of zeros) produce a zero determinant, and the determinant depends linearly on each individual column
3. Formulas for the Determinants
- If A is invertible, then PA=LDU and det P=+1. The product rule gives
(productof pivots)
The sign ±1 depends on whether the number of row exchanges is even or odd. The triangular factors have det L=det U =1 and det D=d1...dn - The determinant of A is a combination of any row i times its cofactors:
det A by cofactors:
The cofactoris the determinant of
with the correct sign:
delete row i and column j
These formulas express detA as a combination of determinants of order n-1
4. Applications of Determinants
4.1 Computation of
-
Cofactor matrix, C is transposed
means
4.2 The solution of Ax=b: Cramer's rule
The jth component of is the ratio
where (has b in column j)
4.3 The Volume of a Box
The determinant equals the volume

4.4 A Formula for the Pivots
- If A is factored into LDU, the upper left corners satisfy
. For every k, the submatrix
is going through a Gaussian elimination of its own.
-
Formula for pivots:
(By convention,
)
Mutiplying together all the individual pivots, we recover:
The pivot entries are all nonzero whenever the numberare all nonzero
- Elimination can be completed without row exchanges (so P=I and A = LU), if and only if the leading submatrices
are non singular
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