Math 2260: Calculus II For Science And Engineering
Spring 2014
How to Compute 2x2 and 3x3 Determinants (for Cross Products)
This document reviews how to compute determinants of 2x2 and 3x3 matrices (i.e. tables
of real numbers which have either 2 rows and columns or 3 rows and columns, respectively).
In this class, we will mostly only be using matrices as a technical tool for working out cross
−
−
products →
u ×→
v more easily. However, matrices have lots of uses in a wide variety of
mathematical disciplines, so I hope you go on to a course like MATH 3000 which covers
them more extensively.
Notation: We write matrices with big square brackets surrounding the cells, and we
typically use capital letters to stand for the whole matrix, like
a11 a12
A=
a21 a22
(Here, the cells have two subscripts: one for their row index, and one for their column index.)
The determinant is a number associated with the matrix, and we write it as either det(A)
or as |A| using absolute-value notation. Thus,
a11 a12 det(A) = a21 a22 are two notations for the same thing. Determinants have lots of uses in geometry, actually,
but we’ll only make use of them in this class for cross products.
2x2 Determinants
If A is a 2x2 matrix, its determinant is
a11 a12
det
= a11 a22 − a21 a12
a21 a22
Think of this as multiplying the elements from the two diagonals (a11 and a22 from the “going
down and right” diagonal, and a21 and a12 from the “going up and right” diagonal), and you
subtract the “up-right” diagonal product from the “down-right” diagonal product.
Samples:
det
det
det
3 2
1 4
0
1
−2 −5
−2 −4
1
2
= 3 · 4 − 2 · 1 = 12 − 2 = 10
= (0)(−5) − (−2)(1) = 0 + 2 = 2
= (−2)(2) − (1)(−4) = −4 + 4 = 0
1
Minors of a 3x3 Matrix
In general, for any matrix with size at least 2x2, if you take any location (i, j) from it
(so the row is i and column is j), the submatrix with (i, j) removed is the matrix you get by
removing column i and row j. For instance, this 3x3 matrix


1
2
3
A =  −1 −2 −3 
4
5
6
has the following three submatrices that come from removing the top row:
−2 −3
−1 −3
−1 −2
For (1, 1):
For (1, 2):
For (1, 3):
5
6
4
6
4
5
The minor of (i, j), written Mij , is the determinant of that submatrix. For instance, in
the example above,
• M11 = (−2)(6) − (5)(−3) = −12 + 15 = 3
• M12 = (−1)(6) − (4)(−3) = −6 + 12 = 6
• M13 = (−1)(5) − (4)(−2) = −5 + 8 = 3
3x3 Determinants
There are several different ways this can be done; let’s present one way of doing this
which comes in handy a lot with cross-products. The procedure is called top-row expansion
of minors. Let’s say the top row of the matrix has a, b, and c in it. Then
det(A) = aM11 − bM12 + cM13
where the M ’s are the minors from the previous section. In other words, you multiply (cell
value) times (minor of that cell) for each element in the top row, and then you alternate
adding and subtracting these products.1
Sample: Take the matrix A from above. Its top row is 1, 2, 3, and we also computed its
minors. Thus,
det(A) = (1)M11 − (2)M12 + (3)M13 = (1)(3) − (2)(6) + (3)(3) = 3 − 12 + 9 = 0
Finding Cross Product via Determinant
−
−
If you have two vectors →
u = hu1 , u2 , u3 i and →
v =
→
−
→
−
formula for u × v is

i j
→
−
→
−

u × v = det u1 u2
v1 v2
1
hv1 , v2 , v3 i in 3D, the determinant

k
u3 
v3
If you read something about linear algebra, you’ll hear the product of ±(cell)(minor) referred to as a
cofactor. We will not generally use that terminology.
2
Basically, you put the first vector’s coordinates along the middle and the second vector’s
coordinates along the bottom. When you do an expansion of minors, you get M11 i − M12 j +
M13 k.
−
−
Sample: If →
u = h−1, −2, −3i and →
v = h4, 5, 6i, then the matrix


i
j
k
 −1 −2 −3 
4
5
6
has the same minors as the examples from earlier: M11 = 3, M12 = 6, and M13 = 3.
Therefore,
→
−
−
u ×→
v = 3i − 6j + 3k = h3, −6, 3i
Sample: h1, 0, 2i × h−1, 1, 5i =
i j k 1 0 2 (determinant)
−1 1 5 1 2 1 0 0 2 (top-row expansion)
− j
+ k
= i
1 5 −1 5 −1 1 = i(0 · 5 − 2 · 1) − j(1 · 5 − (−1) · 2) + k(1 · 1 − (−1) · 0) (2x2 formulas)
= (−2)i − (2)j + (1)k = h−2, −2, 1i
3