This document provides examples and exercises on calculating determinants of matrices using different methods. Example A shows calculating the determinant of a 2x2 matrix by taking the product of diagonal elements and subtracting the product of off-diagonal elements. Examples B-D demonstrate calculating determinants of 3x3 matrices using row expansion, butterfly method, and cofactor expansion. Exercise A gives matrices and asks to calculate their determinants. Exercise B asks to find the characteristic polynomials of matrices in Exercise A. Exercise C provides optional inductive proofs of properties of determinants.

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

3. Determinant
Exercise A.
Calculate the determinants of the following matrices.
1 0
0 1
1) 2) 3) 4)1 5
0 1
a 5
0 𝑏
0 𝑏
𝑎 5
3 5
–2 1
5) 6) 7) 8)
1 –2
7 –1
𝑎 𝑏
𝑏 𝑎
𝑎 𝑏
𝑎 𝑏
9) 10) 11) 12)
1 0 0
0 1 0
0 0 1
1 2 3
0 1 4
0 0 1
a 0 0
0 b 0
0 0 c
a 2 3
0 b 4
0 0 c
13) 14) 15) 16)
1 0 2
0 1 0
−1 0 1
1 2 3
0 1 4
−1 2 1
2 0 −1 0
1 0 0 2
0 1 1 0
−1 2 0 1
2 0 −1 0
1 0 0 2
0 1 1 0
−1 2 0 1

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...)

6. (Answers to the odd problems) Exercise A.
1) det(𝐴) = 1
Exercise B.
3) 𝑥2 − (𝑎 + 𝑏)𝑥 + 𝑎𝑏1) 𝑥2 − 2𝑥 + 1
3) det(𝐴) = 𝑎𝑏
5) 𝑥2 − 4𝑥 + 13
5) det(𝐴) = 13
7) 𝑥2 − 2𝑎𝑥 + 𝑎2 − 𝑏2
7) det 𝐴 = 𝑎2
− 𝑏2 9) det(𝐴) = 1
9) −𝑥3 + 3𝑥2 − 3𝑥 + 1
11) det(𝐴) = 𝑎𝑏𝑐
11) 𝑐 − 𝑥 (𝑥2−𝑎𝑥 − 𝑏𝑥 + 𝑎𝑏)
13) det(𝐴) = 𝑎𝑏𝑐
13) −𝑥3 + 3𝑥2 − 5𝑥 + 3
15) det 𝐴 = −11
15) 𝑥4 − 4𝑥3 + 𝑥2 + 11𝑥 − 11
Determinant