Determinants

A determinant is a single numerical value that is calculated from a square matrix. It determines the properties of a matrix and is computed by adding the products of elements and their corresponding cofactors.

Every matrix has a unique determinant.

Determinants act as a scaling factor for matrices, i.e., they represent how a matrix stretches or compresses space. They are denoted as det(D) or D.

Let's practice this using a problem.

Practice problem: Consider the matrix:

Its determinant is shown as:

Different methods are used for calculating square matrices of different orders.

 

• A square matrix of order 1 × 1 has only one number, so its determinant is the number itself.
  
  
   
• For a square matrix of order 2 × 2, the determinant can be calculated using the determinant formula: 
  
  
   

For example: For the given 2 × 2 matrix


The determinant will be,
\(det(A) = (45)-(32)=20-6=14\)

 

• For a 3 × 3 matrix, 
  
  
  The determinant of a 3 × 3 matrix is calculated using the following steps:
   

1. Take a₁ as the reference element. Now, remove the row and column containing a₁ from the matrix. This gives you a 2 × 2 minor matrix.
   
    
2. Similarly, calculate the minors of b₁ and c₁                              
    
   
    
3. Multiply each number by its minor and sign based on position.
    
4. Sum them all together.
   
    

Let’s take a 3 × 3 matrix:


The values are: a₁ = 1, b₁ = 2, c₁ = 3

1. Minor of a₁ = 1: 
   
   \((4) (6) - (5) (0) = 19\)
    
2. Minor of b₁ = 2: 
   
   \((0) (6) - (5) (1) = 1\)
    
3. Minor of c₁ = 3: 
   
   \((0) (0) - (4) (1) = - 4\)
    
4. Now, apply signs and multiply
   \(det (A) = +1(19)-2(1) +3(-4)\\   \quad \quad   =   19+(-2-12)= 19-14=5\)

• For a 4 × 4 matrix:
  
  
  A few things to keep in mind while solving for the determinant of any 4 × 4 matrices are:

1. Start by multiplying +a₁​ by the determinant of the 3 × 3 matrix, formed by removing the row and column that contain a_​1.
    
2. After doing so, subtract -b₁ multiplied by the determinant of the 3 × 3 matrices formed after removing the row and column containing b₁,
    
3. Next, add +c₁​ times the determinant of the 3 × 3 matrix obtained by removing the row and column containing c_1​.
   
    
4. Finally, subtract -d₁​ multiplied by the determinant of the 3 × 3 matrix formed by deleting the row and column containing d₁​.
    
5. Now, the determinant can be calculated in the same way as that of a 3 × 3 matrix

Parent Tip: Encourage your child to practice finding determinants first by using 2 × 2 matrices, and then move to 3 × 3 and 4 × 4.

Determinants are calculated based on their properties. These properties are:

• Property 1: Identity matrices always have a determinant = 1 
  
  For example, consider a 3 × 3 identity matrix
  
  Then, \(I = 1 \times 1 \times 1 = 1  \)                   

• Property 2: Square matrices having a zero row or column have a determinant value of zero.
  
  For example, take a matrix 
  
  \(  B = (0)(5)-(2)(0) =0 \)

• Property 3: For any triangular matrix (upper or lower), the determinant is calculated by multiplying all elements present on its main diagonal. 
  
  Let's consider a lower triangular matrix:
  
  \(\begin{bmatrix} 6 &0&0 \\2 &5&0\\1&-3&2 \end{bmatrix}\)
  
  \(|c| = 6 \times 5 \times 2 = 60\)
   

• Property 4: For a square matrix, if a row is multiplied by a constant k, then k is a factor of the determinant.
  
  For example, let's take matrix:
                                          
  \(D = \begin{bmatrix} a & b \\ c&d\end{bmatrix}\) 
  
  \(|D| = ad - bc\)

1. Now, we will multiply the first row by k:
   
   \(D' = \begin{bmatrix} ka & kb \\ c&d\end{bmatrix}\)         
   
   Then, \(D'= (ka)(d)-(kb)(c)=k(ad-bc)=k\times D\)

    

2. Let’s take actual values to understand this better:   
   
   \(D = \begin{bmatrix} 2 & 1 \\ 3&4\end{bmatrix}\)                                  
   
   \(D = (2)(4)-(1)(3)=8-3=5\)
    

3. Let’s take k = 3, after multiplication
   
   \(D' = \begin{bmatrix} 6 & 3 \\ 3&4\end{bmatrix}\)
   
   \(D' = (6)(4) - (3)(3) = 24 - 9 = 15\\ So, \ 3 \times |D| = 3 \times 5 = 15\)

• Property 5: For two identical rows or columns, the determinant is zero.
  
  When a matrix has two same rows or columns, it becomes linearly dependent. This means that the matrix has no unique solution when treated as a system. So the determinant is 0.

• Property 6: When any two rows and columns are interchanged, the value of the determinant remains unaffected but its sign changes.

• Property 7: For two square matrices A and B of order n × n, det(A × B) = det(A) × det(B). 
  
  This property explains that the determinant of a product of two matrices is equal to the product of their separate determinants.

• Property 8: The relationship between square matrix D and its adjoint adj(D) is given by:
  \(D \cdot adj(D)=(D)\cdot D=D\cdot I\)
  
  If D is a square matrix, and adj(D) is its adjoint, then the product of D and adj(D) is a scalar multiple of the identity matrix I, and the scalar is det(D).

Rules for operations on determinants help perform row and column operations. These rules are as follows:

1. When two rows or columns are interchanged, the sign of the determinant changes.
    
2. If a row is multiplied by a constant K, then the determinant is also multiplied by the same constant.
    
3. Adding or subtracting a multiple of a row or column from another does not change the determinant.
    
4. If any row or column in a matrix consists of all 0 elements, then the determinant is 0.
    
5. If the elements in a single row or column are written as a sum of two terms, then the determinant can also be written as the sum of two separate determinants, each with one of those terms. This is only applicable if other rows and columns remain unchanged.

Some key elements we need to consider while calculating the determinants of a matrix are: minors and cofactors

1. Minor: A minor is a determinant for every element by the elimination of rows and columns of that element. They are required to find determinants of single elements of a matrix.
   
   
   
   Practice Problem: Find the minor of the element 4 in the determinant: 
   
   \(\begin{vmatrix} 3&2&1 \\ 5&4&6\\0&1&2\end{vmatrix}\)
   
   Element 4 is in the second row and second column. To find its minor, we remove the second row and column. This leaves us with:
   
   \(\begin{vmatrix} 3&1\\0&2\end{vmatrix}\)
   
   \((3\cdot 2)-(1 \cdot 0)=6-0=6\)
   
    
2. Cofactors: Cofactors and minors are related by a formula
   \(C_{ij} = (-1)^{i+j} M_{ij}\)
   
   Where C_ij is the cofactor of the element, and M_ij is the minor.
   
   
   
   Practice Problem: Find the cofactor located in the first row, second column of the given determinant:
   
   \(\begin{bmatrix} 3&5&2\\1&4&6\\7&0&1\end{bmatrix}\)
   
   The element is 5. Removing the mentioned row and column, we get:
   
   \((1\cdot 1)-(6\cdot 7) = 1-42=-41\\ So, \ M_{12} =  -41\)
   
   Cofactor C₁₂ = \((-1)^{1+2}  M_{12}= (-1)^3  (-41) = -1 \cdot (-41) = +41\)

   
   The matrix consisting of all cofactors are called cofactor matrix.

During the introduction to the topic, students may find determinants confusing and difficult. So, here are a few tips and tricks to help to master determinants.

1. Remember, the determinant of an identity matrix is always 1.
    
2. To find the determinant of an upper or lower half triangular matrix, just multiply the diagonal elements.
    
3. Remember, if the matrix has identical rows or columns, then its determinant is always zero.
    
4. If any row or column is zero, then |D| = 0.
    
5. For a 2 × 2 matrix, multiply the diagonal elements and subtract the product of non-diagonal elements.

Parent Tip: Help memorize tour child properties of determinant, and encourage them to practice. You can also use a determinant calculator to check and verify your child’s answers.

Determinants play a crucial role in real-world problems relating to fields involving linear systems and transformations. Given below are a few practical applications of determinants across varied domains.

1. Structural analysis in civil engineering:
   Engineers use determinants to analyze forces acting on structures like bridges and buildings. Determinants help determine if the system is solvable or not.
   
    
2. Help in geometric transformation in computer graphics
   Determinants are used to understand whether a transformation will preserve its orientation or flip it during changes. They are also used for calculating area/volume changes during transformations.
   
    
3. Decoding and encoding in cryptography
   A valid decryption only exists if the determinant of the encryption matrix is non-zero and co-prime to the modulus. These properties of determinants are applied to matrix-based ciphers. Like the Hill cipher
   
    
4. Leontief input-output models in economics
   Economists use determinants in Leontief input-output models to study the interactions of different sectors within an economy.
   
    
5. Analyzing electrical circuits using Kirchhoff’s laws
   Determinants help find unknown current or voltage values across various components of a complex circuit, as Kirchhoff’s Current Law and Voltage Law lead to a system of linear equations.