1. Example A. Find the area.
Determinant
(5,–3)
(1,4)
det
5 –3
1 4
To find the area, we calculate
= 5(4) – 1(–3) = 23 = area
Note that if a, b, c and d are integers,
then the -area is an integer also.
Example C. Calculate by expanding the 1st row.
det
1 0
3 –1= 1*det
1 0 2
2 1 0
2 3 –1
det– 0* det+ 2*
2 0
2 –1
2 1
= 1(–1 – 0) – 0 + 2(6 – 2)
= –1 + 8
= 7
2. Determinant
det
1 0 2
2 1 0
2 3 –1
Example D. Calculate using the Butterfly method.
1 0 2
2 1 0
2 3 –1
1 0
2 1
2 3
Subtract 4 0 0
Add –1 0 12
(–1 +12) – 4 = 7
Example E. Find the determinant using
the 1st row expansion.
2 0 −1 0
1 0 0 2
0 1 1 0
−1 2 0 1
det
= 2 det
0 0 2
1 1 0
2 0 1
– 0 + (−1)det
1 0 2
0 1 0
–1 2 1
– 0 = −11
4. Determinant
Given a matrix Anxn and let x be a variable,
det( A – x * In) is a polynomial in x.
then A – x * In
=
1 2
3 4
1 2
3 4
– x 0
0 x
1 − x 2
3 4 − x
=
This polynomial is called the characteristic polynomial of A.
and
det
1 − x 2
3 4 − x
= (1 – x)(4 – x) – 6 = x2 – 5x – 2
Ex. B. Find the characteristic polynomials of the matrices
from problems in Ex. A. Use this symbolic software, (or any
software) to verify your answer.
For example, if A =
5. Determinant
Ex. C. (Optional)
Many properties of determinants are verified using inductive
arguments on the size of the matrix. Start with verifying the
statement for the case n = 2, or for 2x2 matrices.
Then use the 1st row-expansion for determinants to extend the
pattern or property to the next larger size matrices.
1. Use an inductive argument to validate that if an nxn matrix A
has an entire row of 0’s (or a column of 0’s) then det A = 0.
2. Use an induction to validate that if a row (or a column) of A
is multiplied by c, then the new determinant is c(detA).
3. Use induction to validate that if A is nxn matrix,
then det (cA) = cn det (A)
(In 2D, if we double the length of the sides of a parallelogram,
the new area is 22 = 4 times of the size of the original area.
In 3D, if we double the length of the sides of a “tilted box”,
the new volume is 23 = 8 times of the original volume, etc...)