Rules for Calculating Determinants

In the article Definition Determinants have explained how to calculate the matrix determinant for a matrix of 2 x 2 and 3 x 3 using the Sarrus method. In this article we will show how the determinant of a matrix can be calculated using cofactor expansion and elementary row operations.

Calculating Determinants Using Cofactor Expansion

Theorem 1

If A is a quadratic matrix, then the ai minor entry is expressed by Mij and is defined as a fixed submatrix determinant after the row-i and Column-j are crossed from A. Numbers (-1)i+j Mij is expressed by Cij and is called the cofactor entry aij

Example 1

For example, 




a11 's minor entry is

a11 cofactor is


Likewise, a32 minor entry is
a32 cofactor is


A quick way to determine whether the use of the "+" or "-" sign is the fact that the use of a sign that connects Cij and Mij is in the row-i and column-j of the arrangement.


For example, C11 = M11, C21=-M21, C12 = -M12, C22 = M22, and etc.
Calculate the determinant of the general 3 x 3 matrix below, using a Cofactor Expansion


The determinant of this matrix is ​​written as


What can be written into


Because statements in parentheses are nothing but cofactors C11, C12, and C13, we get

Det(A)=a11C11 + a12C12 + a13C13

Equation (**) shows that the determinant of A can be calculated by multiplying the entries in the first row A with the cofactors and increasing the product of the multiplication. This method of calculing det (A) is called a Cofactor Expansion along the first row A.

Example 2.
For example, 


Calculate the determinant of matrix A with the Expansion method along the first row.


By rearranging the terms as in equation (**) in various ways, it will be possible for us to get formulas like the following:


Note that in each equation all cofactor entries come from rows or from the same column. This equation is called the Cofactor expansion expansion det (A).
Theorem:

The determinant of matrix A, which is n x n, can be calculated by multiplying the entries in a row or column with its cofactors and adding the results of multiplication produced, ie for every 1 i n and 1 j n, then

and

Example 3.
Let A be the matrix in example 2. Calculate det (A) using a cofactor expansion along the first column.
Because it uses cofactor expansion along column-1, the formula used is,



Note: formula Cij = (-1)i+j Mij

Cofactor expansion and row operations or column operations can sometimes be used together to provide an effective method for calculating determinants.

Example 4. 
Calculate det (A) note that matrix A is,

Solution:
By adding the corresponding multiplication from the second row to the rest of the line, we get.

Add - (the value of row-i column-1) second row to the other row to get the method as in the example calculation method of Gauss.hitungan Metode Gauss.


With cofactor expansion as long as the first column is obtained,

By adding the first row to the third row it will produce,


With the expansion of the cofactor along the first column it will give,




In cofactor expansion, we calculate det (A) by multiplying entries in rows or columns with their cofactors and adding the resulting multiplication results. It turns out that if we multiply the entries in a row with the corresponding cofactors from different rows, then the sum of these multiplication results is always equal to zero.

Example 5
Suppose we have a matrix A,
Review it



It is equal to zero, by forming a new A' matrix by replacing the third line with a copy of the lian from the first row. So,


Suppose C’31, C'32, and C’33 are cofactors of the entries in the third row of A '. Because the first two lines A and A are the same, and because of calculations C31, C32, C33 and C'32 and C33 only involve the second entries of the first row of A and A, then we get

C31=C’31                                 C32=C’32                                 C33=C’33

Because A' has two identical lines then, conversely by calculating (A') and cofactor expansion along the third line it will give,

So we get it,.

Definition
If A is any n x n and Cij is a cofactor aij, then the matrix


Named a cofactor matrix A. Transpose matrix is ​​called adjoin A and is stated with adj(A).

Example 6. 

Suppose that you calculate the C11 cofactor and other cofactors and arrange them like a matrix

C11=(-1)1+1M11=(-1)2 (a22a33-aa32a23)=1((6x0)-(-4x3))=12

C11 = 12         C12 = 6           C13 = -16
C21 = 4           C22 = 2           C23 = 16
C31 = 12         C32 = -10        C33 = 16

So the cofactor matrix is


While Adjoin A is


Calculating Determinants with Elemental Line Operations

In this section we will show that the determinant of a matrix can be calculated by reducing the matrix in the form of row echelons. This method is important to avoid the length calculations involved in applying the determinant definition directly.

Theorem 2
If A is any quadratic matrix containing a row of zero numbers then det(A) = 0

Proof, because the elementary result marked with A contains one factor from each line A, then each marked elementary product contains a factor of the zero number row and consequently will also have zero. Because det (A) is the sum of all elementary times marked, we get det(A) = 0.

Example 7.
Calculate the determinant of the matrix below,

Solution:

Note,  

We do not need further reductions, because if the matrix of squares has two comparable lines, like the first row and the second row of matrix A in Example 7 above, then the determination is zero.
The next theorem shows how elementary row operations on the matrix will affect the determinant value.

Theorem 3.
Let A be any matrix n x n.
  1. If A' is a matrix that results when two lines A are exchanged, then det (A') = - det (A).
  2. If A' is a matrix that results when a single row A is multiplied by a constant k, then det (A) = k det (A).
  3. If A 'is a matrix that is generated when multiples of one line A are added to another line, then det (A') = det (A).

Example 8.
Calculate the determinant of the Matrix below,


Solution: By reducing A to the shape of the row echelon, then we get


Theorem 4.
If A is a triangular matrix n x n, then det(A) is the product of entries in the main diagonal, namely det(A) = a11a22...a33.


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