Chapter 2: Systems of
Linear Equations and
Inequalities
Determinants and Multiplicative
Inverses of Matrices
Objectives
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Evaluate determinants
Find inverses of matrices
Solve systems of equations using inverses of
matrices
Learn Cramer’s Rule for solving systems of
equations
History of the Determinant
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The concept of the determinant of a matrix
appeared in Europe and Japan at almost
identical times. Seki of Japan wrote about it
in 1683 and in the same year, Leibniz of
Germany also wrote about it in his works.
In 1850 the word matrix was first used by
James Sylvester to describe the tabular array
of numbers. Today computer experts use
matrices to solve problems that involve
thousands of numbers.
Second and Third-Order
Determinants
Each square matrix has a determinant. A matrix can NOT have a determinant
unless it is a square matrix (same number of rows and columns). The
determinant in NOT a matrix. It is a number associated with a matrix.
A matrix that has a nonzero determinant is called nonsingular.
Second-order determinant Value of the det. of
Third-order determinant Value of the det. of
a1
b 2 c2
b 3 c3
- b1
a2 c2
a3 c3
+ c1
a2 b2
a3 b3
a 1 b1
a 2 b2
= a1b2 - a2b1
a1 b1 c1
a2 b2 c2
a3 b3 c 3
=
Finding the values of a matrix
means finding the determinant
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Find the value of ⎡0 - 2⎤
⎢8 - 6 ⎥
⎣
⎦
The value (or determinant) is 0(-6)-8(-2)=16
Find the value of ⎡5 3 - 1⎤
⎢ 6 4 8⎥
⎣
⎦
4 8
6 8
6 4
The value (or determinant) is 5 -3 7 -3 0 7 -1 0 3 =
5 4(7)- -3(8) -3 6(7)-0(8) -1 6(-3)-0(4)
260-126+18=152
= 5(52) -3(42) -1(-18)=
Identity Matrix for
Multiplication
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The identity matrix for any square matrix A is the
matrix I such that IA=A
The identity matrix for multiplication for any secondorder matrix is ⎡1 0⎤
⎢0 1⎥
⎦
⎣
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The identity matrix for any third-order matrix is
⎡1 0 0⎤
⎢0 1 0 ⎥
⎢
⎥
⎢⎣0 0 1⎥⎦
Inverse Matrix
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A
−1 is the multiplicative inverse of A.
AA −1 = 1
⎡a1 b1 ⎤
Suppose A is equal to the matrix ⎢a b ⎥ and the inverse matrix
⎣ 2 2⎦
x
y
⎡
⎤
1
1
can be designated as
⎢x y ⎥
⎣ 2 2⎦
−1
The product of the matrix A and its inverse
identity matrix ⎡1 0⎤ .
⎢0 1⎥
⎣ ⎦
A A −1 =
⎡1 0⎤
⎢0 1⎥
⎣ ⎦
or
A
must equal the
⎡a1 b1 ⎤ ⎡ x1 y1 ⎤
=
⎢a b ⎥ ⎢
⎥
⎣ 2 2 ⎦ ⎣x 2 y 2 ⎦
Note if a matrix A has a determinant of 0, then
A−1
A
−1
⎡1 0⎤
⎢0 1⎥
⎣ ⎦
does not exist.
Inverse of a second order
matrix
If
⎡a1 b1 ⎤
⎡a1 b1 ⎤
A=⎢
⎥ ≠ 0,
⎥ and ⎢
⎢⎣a 2 b 2 ⎥⎦
⎢⎣a 2 b 2 ⎥⎦
⎡b 2 - b1 ⎤
1
then A−1 =
⎥
⎢
a
a
⎡a1 b1 ⎤ ⎢⎣ 2 1⎥⎦
⎥
⎢
⎢⎣a 2 b 2 ⎥⎦
We will now use this information to learn one way
to solve systems of linear equations using
matrix equations.
Solving systems of equations
using matrix equations
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We know that if
AX
= B
−1
−1
, then A AX = A B
X = A−1 B
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We can use this information to solve a system of linear equations.
Solve the system of equations by using matrix equations.
4x-2y=16
x+6y=17
⎡4 - 2⎤ ⎡ x ⎤ ⎡16 ⎤
Write it as a matrix equation ⎢⎢1 6 ⎥⎥ ⋅ ⎢⎢ y ⎥⎥ = ⎢⎢17 ⎥⎥
⎣
Find the inverse of
⎦ ⎣ ⎦
⎡ 4 - 2⎤
1
⎢
⎥=
⎢⎣1 6 ⎥⎦ ⎡4 - 2⎤
⎢
⎥
⎢⎣1 6 ⎥⎦
1
4(6) − 1(−2)
1
26
⎡6 2 ⎤
⎢
⎥
⎢⎣- 1 4 ⎥⎦
⎣
⎦
⎡6 2 ⎤
⎢
⎥=
⎢⎣- 1 4⎥⎦
⎡6 2 ⎤
⎢
⎥=
1
4
⎢⎣
⎥⎦
Continuation
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Now multiply each side of the matrix equation by the
inverse and solve.
⎡6
⎢
⎢⎣− 1
1 ⎡6
⎢
26 ⎢⎣- 1
1
26
⎡x ⎤
1 ⎡6(16) + 2(17) ⎤
⎢ ⎥=
⎢
⎥=
y
1
(
16
)
4
(
17
)
−
+
26
⎢⎣ ⎥⎦
⎢⎣
⎥⎦
1
26
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⎡130⎤ ⎡5 ⎤
⎢ ⎥=⎢ ⎥
⎢⎣52 ⎥⎦ ⎢⎣2⎥⎦
So the solution is (5,2)
2 ⎤ ⎡ 4 - 2⎤ ⎡ x ⎤
⎥⋅⎢ ⎥ =
⎥⋅⎢
4⎥⎦ ⎢⎣1 6 ⎥⎦ ⎢⎣ y ⎥⎦
2⎤ ⎡16 ⎤
⎥⋅⎢ ⎥
4 ⎥⎦ ⎢⎣17 ⎥⎦
Cramer’s Rule
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Gabriel Cramer developed an algorithm to solve
systems of linear equations.
Ax + By=C
AF − CD
CE - BF
Dx + Ey=F
Then x = AE - BD
y = AE − BD
Using matrices x =
⎡C B ⎤
⎢F E ⎥
⎣
⎦
⎡ A B⎤
⎢D E ⎥
⎣
⎦
y=
⎡A C⎤
⎢D F ⎥
⎣
⎦
⎡ A B⎤
⎢D E ⎥
⎣
⎦
Use Cramer’s Rule
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Solve 6x-3y=63
5x-9y=85
63(−9) − (−3)(85)
=
6(−9) − (−3)(5)
− 312
=8
− 39
6(85) − 63(5)
=
6(−9) − (−3)(5)
195
= −5
− 39
z
X=
z
Y=
z
So the solution to the system of equations is (8,-5)
HW #15
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Section 2-5
Pp 102-105
#15-23 odd, 34,35, 36 (Use Cramer’s Rule)