# 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.