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251
Instruction: Introduction to Matrices & Determinants
Previous lectures have focused on functions whose inputs and outputs were real numbers. In
other words, the previous lectures discussed functions of the form f : X 6 Y where X = Y = \ . This
lecture discusses functions whose inputs are square matrices.
A matrix is an array of real numbers. Matrix B below is a 3×4 matrix, that is, a matrix with
three rows and four columns. The numbers arrayed in a matrix are called entries, denoted aij , the
entry in the ith row and jth column. The entry in the second row, third column of matrix B below is
thirteen.
4
5
−2⎤
⎡2
⎢
B = ⎢1 − 1 13
7 8⎥⎥
⎢⎣ 6
9 23
10 ⎥⎦
A square matrix has as many rows as columns. Matrix A below is a square 2×2 matrix.
⎡4
A=⎢
⎣0
− 3⎤
2 ⎥⎦
Determinants are functions whose inputs are square matrices and whose outputs are real
numbers. Let M be the set of square matrices with real number entries. Let Y = \ . A determinant is a
function of the form f : M → Y . This lecture will deal exclusively with determinants of 2×2 matrices.
a ⎤
a
a
⎡a
Let A = ⎢ 11 12 ⎥ . The determinant of A, denoted either det ( A ) or 11 12 , is
a21 a22
⎣ a21 a22 ⎦
the function f such that f : A 6 a11 ⋅ a22 − a21 ⋅ a12 .
Note that the use of A signifies a determinant, not an absolute value. According to the definition
0⎤
⎡a b ⎤
⎡2
and S = ⎢
above, if T = ⎢
⎥
⎥ , then det (T ) = ad − bc , and S = −8 .
⎣c d ⎦
⎣0 − 4 ⎦
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Instruction: Determinants of 3 × 3 Matrices
If a matrix is a square matrix (the number of rows is equal to the number of columns) as
above, then there is a number associated with the matrix called its determinant. Previously, we
admitted that determinants are mappings from square matrices with real number entries to real
numbers.
Placing special emphasis on the output of the function, a determinant is a real number
associated with a square matrix and is indicated by enclosing the array between two vertical bars.
For a matrix A, the corresponding determinant is designated as det(A) and is read "determinant of
A." For example,
⎡1 2 ⎤
matrix A = ⎢
⎥
⎣3 4 ⎦
det ( A ) =
1 2
3 4
A determinant may arise from any n × n matrix where n is a positive integer greater than
one. This discussion will be 3× 3 determinants.
Every square matrix with real entries has a determinant. The method for finding the
value of 3× 3 determinants involves finding the value of 2 × 2 determinants. Determinants of
larger matrices can be evaluated by using techniques similar to those shown here.
The value of a 2 × 2 determinant is the difference between the products of the diagonals
as given by the following definition:
a b
⎡a b ⎤
For the square matrix A = ⎢
, det(A) =
= ad − cb.
⎥
c d
⎣c d ⎦
One method of evaluating 3× 3 determinants is called expanding by minors. In
this method, one row (or column) is chosen and each element in that row has a minor.
Each minor is found by mentally crossing out both the row and column (shown below
with dotted lines) that contain that element. The minors of the elements of the first row
are shown here:
c11 c12 c13
c 22 c 23
c 21 c 22 c 23
→
← minor of c11
c32 c33
c31 c32 c33
c11
c12
c13
c 21
c 22
c 23
c31
c32
c33
c11
c12
c13
c 21
c 22
c 23
c31
c32
c33
→
c 21
c 23
c31
c33
→
c 21
c 22
c31
c32
← minor of c12
←
minor of c13
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Finding the product of each element and its minor and then adding these products with an
adjustment of alternating signs finds the value of a determinant as given by the following
definition:
c11 c12 c13
⎡c11 c12 c13 ⎤
⎢
⎥
A = ⎢c 21 c 22 c 23 ⎥, det( A) = c 21 c 22 c 23
⎢⎣c31 c32 c33 ⎥⎦
c31 c32 c33
c 22 c 23
c 21 c 23
c
c 22
= c11 ⋅
− c12 ⋅
+ c13 ⋅ 21
c32 c33
c31 c33
c31 c32
The value of the determinant may also be found by expanding by the minors of any other
row or any column. Each minor is the product of its corresponding element and –1 to the r + c
power, (–1)r+c, where r is the row and c is the column of the element. The factor (–1)r+c always
produces the following pattern of signs:
+ − +
− + −
+ − +
For instance, the value of a 3× 3 determinant could be found by expanding by the minors of row
two according to the chart above: det(A) = –c21(minor of c21) + c22(minor of c22) – c23(minor of
c23).
Consider matrix C.
1
− 4⎤
⎡5
⎢
C = ⎢2
6
3 ⎥⎥
⎢⎣ 2
2
1 ⎥⎦
The determinant of C is found below by expanding by the minors of row one.
det (C ) = 5
6 3
2 3
2 6
−1
+ (− 4)
2 1
2 1
2 2
det (C ) = 5(6 ⋅ 1 − 2 ⋅ 3) − 1(2 ⋅ 1 − 2 ⋅ 3) − 4(2 ⋅ 2 − 2 ⋅ 6)
det (C ) = 5(6 − 6 ) − 1(2 − 6) − 4(4 − 12 )
det (C ) = 5(0 ) − 1(− 4 ) − 4(− 8)
det (C ) = 36
In the next example, the process for finding a 3× 3 determinant is simplified by
transforming the matrix with elementary row and column operations first.
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⎡6
⎢− 2
⎢
⎢⎣− 3
⎡0
⎢− 2
⎢
⎢⎣− 3
⎡6 − 2 4 ⎤
D = ⎢⎢− 2 9 − 1⎥⎥
⎢⎣− 3 2 − 1⎥⎦
− 2 4⎤
⎡ 0 25 1 ⎤
⎥
9 − 1⎥3 ⋅ R2 + R1 → R1 ⎢⎢− 2 9 − 1⎥⎥
2 − 1 ⎦⎥
⎣⎢− 3 2 − 1⎥⎦
25 1 ⎤
⎡0 0 1 ⎤
⎥
9 − 1⎥ − 25C 3 + C 2 → C 2 ⎢⎢− 2 34 − 1 ⎥⎥
⎢⎣− 3 27 − 1⎥⎦
2 − 1⎥⎦
⎡ 6 − 2 4⎤
⎢ − 2 9 − 1⎥
⎥
⎢
⎢⎣− 3 2 − 1 ⎥⎦
↔
row-column
equivalent
1⎤
⎡0 0
⎢− 2 34 − 1⎥
⎥
⎢
⎢⎣− 3 27 − 1⎥⎦
6 −2 4
det( D) = − 2 9 − 1
− 3 2 −1
0 0
1
det( D) = − 2 34 − 1
− 3 27 − 1
det( D) = 0
− 2 − 1 − 2 34
34 − 1
−0
+1
− 3 −1
− 3 27
27 − 1
det( D) = 1(− 2 ⋅ 27 − (− 3) ⋅ 34 )
det( D) = −54 + 102
det( D) = 48
Instruction: Some Applications of Determinants
It can be shown that the area of a triangle formed by the points (x1,y1), (x2,y2), (x3,y3) is
x1 y1 1
1
given by the absolute value of
x 2 y 2 1.
2
x3 y 3 1
For example, consider the triangle determined by (0,0),
(0,6), (4,0) as shown here.
(0,6)
(0,0)
(4,0)
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Recall that the area of a triangle equals the product of its base, height, and one-half.
Accordingly, the area of the triangle is ½× 4 × 6 = 12. Substituting the coordinates of the four
vertices of the triangle into the determinant formula yields the same area:
0 0 1
1
1
1
1
Area = ⋅ ABS 0 6 1 = ⋅ ABS [4(0 − 6)] = ⋅ ABS (− 24 ) = ⋅ 24 = 12.
2
2
2
2
4 0 1
(Note the use of ABS for absolute value. The determinant formula presents some notational
difficulties since the symbol for absolute value, two vertical segments, is appropriated to indicate
the determinant.)
The determinant formula can also be used to show that three points are collinear.
x1 y1 1
1
Suppose one calculates the absolute value of
x 2 y 2 1 and gets zero. This means the area
2
x3 y 3 1
of the triangle is zero. If the area of a triangle is zero, then the three points must lie in a line.
Thus, the determinant formula for the area of a triangle provides a check for collinearity. Three
x1 y1 1
1
points, (x1,y1), (x2,y2), (x3,y3), are collinear if and only if the absolute value of
x 2 y 2 1 = 0.
2
x3 y 3 1
Determinants present an alternate method for finding the equation of a line through two
points. In a Cartesian plane, lines have equations of the form Ax + By + C = 0, and any equation
of the form Ax + By + C = 0 is a line. Consider the equation attained from the determinant on the
next page.
x
y 1
x1 y1 1 = 0
x2
x
y2
y1 1
y2
1
1
−y
x1 1
x2
1
+1
x1
y1
x2
y2
=0
x ( y1 − y2 ) − y ( x1 − x2 ) + ( x1 y2 − x2 y1 ) = 0
For fixed x1, x2, y1, y2 this equation is an equation of the form Ax + By + C = 0, i.e., a line.
Furthermore, this line contains the points (x1,y1) and (x2,y2). To check, let x = x1 and y = y2:
x1 ( y1 − y 2 ) − y1 ( x1 − x 2 ) + ( x1 y 2 − x 2 y1 ) = 0
x1 y1 − x1 y 2 − x1 y1 + x 2 y1 + x1 y 2 − x 2 y1 = 0
x1 y1 − x1 y1 − x1 y 2 + x1 y 2 + x2 y1 − x 2 y = 0
0=0
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Thus, determinants present an alternate method for finding the equation of a line through two
given points. For example, to find the equation of the line that passes through (1,5) and (3,2),
x
y 1
substitute into the equation x1 y1 1 = 0 as shown below.
x2
y2
1
x
y 1
1
5 1 =0
3
2 1
x
5 1
1 1
1 5
−y
+1
=0
2 1
3 1
3 2
x ( 5 − 2 ) − y (1 − 3) + ( 2 − 15 ) = 0
x ( 3) − y ( −2 ) + ( −13) = 0
3x + 2 y − 13 = 0
It is not important to expand using the first row. Any row or column will work.
257
Instruction: Determinants
Example 1
Finding the Determinant of a 2×2 Matrix
1⎤
⎡
−5 − ⎥
⎢
Given B =
2 , find det ( B ) .
⎢
⎥
3⎦
⎣ 6
det( B) =
1
2
3
−5 −
6
⎛ 1⎞
det( B ) = −5(3) − 6⎜ − ⎟
⎝ 2⎠
det( B) = −15 + 3
det( B) = −12.
Example 2
Finding the Missing Entries of a 2×2 Matrix Given the Determinant
⎡ x − 3⎤
Suppose A = ⎢
⎥ and det ( A ) = 8 . What are the possible values of x ?
⎣4 8 ⎦
det( A) =
x −3
4 8
8 = x ⋅ 8 − 4 ( −3)
8 = 8 x + 12
−4 = 8 x
4
− =x
8
1
− =x
2
258
Instruction: Determinants
Example 1
Finding the Determinant of a 3×3 Matrix
⎡6
Given C = ⎢⎢ 2
⎢⎣ 2
2
1
3
− 1⎤
0 ⎥⎥ , find det ( C ) .
9 ⎥⎦
Since row two has a zero and two positive elements, expanding by the minors of row two is
convenient. Recall the sign pattern for row two (–,+,–).
det ( C ) = −2
2 −1
6 −1
6 2
+1
−0
3 9
2 9
2 3
Calculate the determinant of each minor and simplify.
det ( C ) = −2 ( 2 ⋅ 9 − 3 ⋅ −1) + 1( 6 ⋅ 9 − 2 ⋅ −1) − 0 ( 6 ⋅ 3 − 2 ⋅ 2 )
det ( C ) = −2 (18 + 3) + 1( 54 + 2 ) − 0 (18 − 4 )
det ( C ) = −2 ( 21) + 1( 56 ) − 0 (14 )
det ( C ) = −42 + 56 − 0
det ( C ) = 14
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Example 2
Using the Determinant of a 3×3 Matrix to Calculate Area of a Triangle
Suppose a triangle has vertices at ( 0, 0 ) , ( 6, −2 ) , and (1, −8 ) . Find the area of the triangle
using determinants.
⎛ 0 0 1⎞
1
⎜
⎟
Area = ABS ⎜ 6 − 2 1 ⎟
2
⎜ 1 −8 1 ⎟
⎝
⎠
⎛ −2 1
6 1
6 −2 ⎞
1
Area = ABS ⎜ 0 ⋅
− 0⋅
+1
⎟
1 1
1 −8 ⎠
2
⎝ −8 1
1
Area = ABS 0 − 0 + 1( 6 ⋅ ( −8 ) − 1 ⋅ ( −2 ) )
2
1
Area = ABS ( −48 + 2 )
2
1
Area = ABS ( −46 )
2
1
Area = ⋅ 46
2
Area = 23 square units
(
)
260
Example 3
Using the Determinant of a 3×3 Matrix to Find the Equation of a Line
Suppose a line contains the points ( −1, 0 ) and ( 6, 2 ) . Find the equation describing the line
using determinants.
x y 1
−1 0 1 = 0
6 2 1
x⋅
0 1
−1 1
−1 0
− y⋅
+ 1⋅
=0
2 1
6 1
6 2
x ⋅ ( 0 ⋅1 − 2 ⋅1) − y ⋅ ( −1 ⋅1 − 6 ⋅1) + 1 ⋅ ( −1 ⋅ 2 − 6 ⋅ 0 ) = 0
( 0 − 2 ) x − ( −1 − 6 ) y + ( −2 − 0 ) = 0
−2 x − ( −7 ) y + ( −2 ) = 0
−2 x + 7 y − 2 = 0
y=
2
2
x+
7
7
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Problems
#1
⎡5
Given M = ⎢
⎣ −9
#2
Evaluate
#3
⎡x
Let N be a 2 × 2 matrix. Use the function det ( N ) to map ⎢
⎣3
#4
⎡10 x
If V = ⎢
⎢ −2
⎣
#1 det ( M ) = 33
#4 x = 22
2⎤
, evaluate det ( M ) .
3 ⎥⎦
45 − 2
.
19 − 1
1⎤
.
2 ⎥⎦
x⎤
1 ⎥⎥ , find the values of x such that det (V ) = 154 .
2⎦
#2
45 − 2
= −7
19 − 1
⎡x
#3 ⎢
⎣3
1⎤
6 2x − 3
2 ⎥⎦
262
Problems
#1
2
4
Evaluate − 1 3
5 −2
6
3 .
2
#2
Show with any example that the determinant of a matrix is zero if every element in a row or
column of that matrix is zero.
#3
Use determinants to find the area of a triangle formed by (–4, –3), (0,5), (7,1).
#4
Use determinants to find the equation of a line passing through (0,6) and (–3, –8).
1 2
#5
3
What is the value of x if 4 0 − 1 = 6 ?
x 1 −2
ANSWERS
#1 14
0 0 0
1 3
7 3
7 1
− 0⋅
+ 0⋅
=0
1 3 = 0⋅
2 5
4 5
4 2
4 2 5
#2 Consider 7
#3 Area equals 36 square units.
#4 14x – 3y + 18 = 0
#5 x = 11.5
263
Suggested Homework from Blitzer
Section 6.5:
#1-7 odd, #27-31 odd, #51-57 odd
Application Exercise
Surveyors map the counter-clockwise vertices ( x1 , y1 ) , ( x2 , y2 ) ,… , ( xn , yn ) of a
polygonal plot of land to the plot's square area using the function below.
A : ( x1 , y1 ) , ( x2 , y2 ) ,… , ( xn , yn )
1 ⎧⎪ x1 x2 x2 x3
+
+
⎨
2 ⎩⎪ y 1 y2 y2 y3
+
xn x1 ⎫⎪
⎬
y n y1 ⎭⎪
Find the area for a plot of land with vertices ( 9 ft , −2 ft ) , ( 24 ft , 4 ft ) , (18 ft ,12 ft ) ,
and (1 ft , 2 ft ) .