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In this section, you will:
  • Evaluate  2 × 2  determinants.
  • Use Cramer’s Rule to solve a system of equations in two variables.
  • Evaluate  3 × 3  determinants.
  • Use Cramer’s Rule to solve a system of three equations in three variables.
  • Know the properties of determinants.

We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.

Evaluating the determinant of a 2×2 matrix

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.

Find the determinant of a 2 × 2 matrix

The determinant    of a 2   ×   2 matrix, given

A = [ a b c d ]

is defined as

Notice the change in notation. There are several ways to indicate the determinant, including det ( A ) and replacing the brackets in a matrix with straight lines, | A | .

Finding the determinant of a 2 × 2 matrix

Find the determinant of the given matrix.

A = [ 5 2 6 3 ]
det ( A ) = | 5 2 6 3 | = 5 ( 3 ) ( −6 ) ( 2 ) = 27
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using cramer’s rule to solve a system of two equations in two variables

We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule    , this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques . Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.

Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.

To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.

a 1 x + b 1 y = c 1 ( 1 ) a 2 x + b 2 y = c 2 ( 2 )

Questions & Answers

answer and questions in exercise 11.2 sums
Yp Reply
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Alaba
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Chris
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Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
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C'est comment
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what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
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Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
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humberto
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Yvonne
because x is 6
SAIDI
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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