Determinants in Algebra
These is my guide to determinants in algebra.
Introduction
Determinants are used in mathematics for theoretical purposes. They can be used
to test if a matrix is invertible and to find the area of certain geometric
figures. A determinant is a real number associated with a square matrix.
The definition of a matrix with dimension 2*2, is a calculation of:
\[det-A = ad - cb\]
When you have a matrix of:
\[\begin{pmatrix}a&b\\c&d\end{pmatrix}\]
A matrix is invertible if det A is not equal to 0.
Example 1
Determine if \(A^{-1}\) exists by computing the determinant of the matrix A.
\[A=\begin{pmatrix}3&-4\\-5&9\end{pmatrix}\]
Solution:
The determinant of the 2*2 matrix A is calculated as follows:
\[det-A=det\begin{pmatrix}3&-4\\-5&9\end{pmatrix} = (3)(9)-(-5)(-4)=7\]
Since det A does not equal 0, the matrix A is invertible and \(A^{-1}\) exists
Example 2
Determine if \(A^{-1}\) exists by computing the determinant
\[B=\begin{pmatrix}52&-32\\65&-40\end{pmatrix}\]
Solution:
\[det-B=\begin{pmatrix}52&-32\\65&-40\end{pmatrix}=(52)(-40)-(65)(-32)=0\]
since B=0, \(A^{-1}\) does not exist.
We can use determinants of 2*2 matrices to find determinants or larger square
matrices. In order to do this, we first define the concepts of a minor and a
cofactor.
Minors and Cofactors
The minor, denoted by \(M_{ij}\), for element \(a_{ij}\) in the square matrix A
is the real number computed by performing the following steps.
Step 1: Delete the ith row and jth column from the matrix A
Step 2: \(M_{ij}\) is equal to the determinant of the resulting matrix
The cofactor, denoted by \(A_{ij}\) for \(a_{ij}\) is defined by
\(A_{ij}=(-1)^{i+j}M_{ij}\)
Example 3
Calculating Minors and Cofactors
Find the following minors and cofactors for the matrix A.
\[A=\begin{pmatrix}2&-3&1\\-2&1&0\\0&-1&4\end{pmatrix}\]
For: M11 and M21
Solution:
To obtain the minor m11, begin by crossing out the first row and first column of
A.
The remaining elements form the 2*2 matrix:
\[\begin{pmatrix}1&0\\-1&4\end{pmatrix}\]
The minor M11 is equal to det B = (1)(4)-(-1)(0)=4.
M21 is found by crossing out the second row and first column of A.
\[\begin{pmatrix}-3&1\\-1&4\end{pmatrix}\]
M21 = det B = (-3)(4) - (1)(-1) = -11
Determinant of a Matrix using Cofactors
To compute the determinant of a 3*3 matrix A, begin by selecting a row or
column. If the second row of A is selected, the elements are a21, a22, and a23.
Then det A = a21A21 + a22A22 + a23A23.
On the other hand, utilizing the elements of a11, a21, a31 in the first column
gives det A = a11A11 + a21A21 + a31A31. Regardless of the row or column
selected, the value of det A is the same. The calculation is easier if some
elements in the selected row or column equal 0.
Example 4
Evaluating the determinant of a 3*3 matrix
\[A = \begin{pmatrix}2&-3&1\\-2&1&0\\0&-1&4\end{pmatrix}\]
Solution:
To find the determinant of A, we can select any row or column. If we begin
expanding about the first column of A, then:
det A = a11A11 + a21A21 + a31A31
In the first column, a11=2, a21=-2, a31=0.
\[det A = a11A11 + a21A21 + a31A31\]
\[= 2(4) + (-2)(11) + (0)(A31) = -14\]
Area of Regions
Determinants may be used to find the area of a triangle. If a triangle has
vertices (a1,b1),(a2,b2) and (a3,c3), the its area is equal to the absolute
value of D, where:
\[D = 1/2det \begin{pmatrix}a1&a2&a3\\b1&b2&b3\\1&1&1\end{pmatrix}\]
If the vertices are entered into the columns of D in a counterclockwise
direction, then D will be positive.
Example 5
Computing the Area of a Parallelogram
Use determinants to calculate the area of the parallelogram.
Solution:
To find the area of the parallelogram, we view the parallelogram as comprising
two triangles. One possible triange has vertices at (0,0), (4,2), and (1,2), and
the other triangle has vertices at (4,2), (5,4), and (1,2). The area of the
parallelogram is equal to the sum of the areas of the two triangles. since these
triangles are congruent, we can calculate the area of one triangle and double
it. The area of one triangle is equal to D, where :
\[D = 1/2det \begin{pmatrix}0&4&1\\0&2&2\\1&1&1\end{pmatrix} = 3\]
Since the vertices were entered in a counterclockwise direction, D is positive.
The area of one triangle is equal to 3. Therefore the area of the parallelogram
is twice this value or 6.
Cramer's Rule
Determinants can be used to solve linear systems by employing a method called
Cramer's rule.
Example 6
Using Cramer's rule to solve a linear system in two variables
\[4x + y = 146\]
\[9x + y = 66\]
Solution:
In this system, a1=4, b1=1, c1=146, a2=9, b2=1, and c2=66. By Cramer's rule, the
solution can be found like this:
\[E=det\begin{pmatrix}c1&b1\\c2&b2\end{pmatrix}\]
\[E=det\begin{pmatrix}146&1\\66&1\end{pmatrix}=(146)(1)-(66)(1)=80\]
\[F=det\begin{pmatrix}a1&c1\\a2&c2\end{pmatrix}\]
\[F=det\begin{pmatrix}4&146\\9&66\end{pmatrix}=(4)(66)-(9)(146)=-1050\]
\[D=det\begin{pmatrix}a1&b1\\a2&b2\end{pmatrix}\]
\[D=det\begin{pmatrix}4&1\\9&1\end{pmatrix}=(4)(1)-(9)(1)=-5\]
The solution is:
\[x=\frac{E}{D} = 80/-5 = -16 \text { and } y = \frac{F}{D} = -1050/-5 = 210\]
Limitations on the Method of Cofactors and Cramer's Rule
Systems of linear equations involving more than two variables can be solved with
Cramer's rule. If a linear system has n equations, then Cramer's rule requires
the computation of n+1 determinants with dimension n*n. Cramer's rule is seldom
employed in real applications because of the substantial number of arithmetic
operations needed to compute determinants of large matrices.
It can be shown that the cofactor method of calculating the determinant of an
n*n matrix, n>2, generally involves more than n! multiplication operations.
In real life applications, it is not uncommon to solve linear systems that
involve thousands of equations.