Cofactor: Minors & Formula

Cofactor of an element of a matrix is the determinant of the matrix that is obtained by eliminating the row and column in the matrix that contains the element and then multiplying by +1 or -1 as per the position of the element. 

• The main purpose of the cofactor matrix is to determine the inverse of a matrix
• It further helps to know the cofactor of a certain row or column which also helps in determining the determinant of the matrix.

\(\begin{bmatrix}5 & 7 & 0 \\[0.3em]2 & 9 & 1 \\[0.3em]4 & 6 & 3 \\[0.3em] \end{bmatrix}\)

• If there is a given matrix, the cofactor of an element in position 5 is obtained by eliminating the 1 row and 1 column 

9 x 3 – 6 x 1 = 21

• Therefore, the minor corresponding to the element 5 in the first row is 21.
• For calculating cofactor, the formula is C_ij = ((-1)^i+j)M_ij

I = 1, j = 1 and M = 21

C_ij = (-1)² 21 = 21

Key Terms: Cofactor matrix, Determinant, Matrices, Minor, Row, Column, Sum, Even, Odd, Scalar

---

What is a Cofactor Matrix?

[Click Here for Sample Questions]

To understand the Cofactor matrix, one needs to understand minors. Minor is the value which is calculated from the square matrix of the determinant.

The cofactor can also be known as signed minor which is denoted by A_yz, and is defined by 

A= (-1)^y+z M, where M stands for Minor of a_yz.

Rules to be followed:

• If the sum of y+z is even, then A_yz = M_yz.
• If the sum of y+z is odd, then A_yz = – M_yz.
• In terms of sign there is a difference between the related minors and cofactors.

---

How to find a cofactor?

[Click Here for Previous Year Questions]

Thе minor of an еlеmеnt in a matrix is thе dеtеrminant of thе submatrix obtainеd by rеmoving thе row and column containing that еlеmеnt. Once done, apply thе sign rulе to determine thе cofactor.

• Thе sign rule states that thе sign of thе cofactor depends on thе position of thе еlеmеnt in thе matrix. 
•  If thе row numbеr and column numbеr of thе еlеmеnt sum up to an even numbеr, thе cofactor has a positivе sign.
• If thе sum is odd, thе cofactor has a nеgativе sign. 
• Cofactor of an element within a matrix is obtained when the minor M_ij of the element is multiplied with (-1)_i+j. 
• The cofactor of the element is denoted by C_ij and the if the minor of the element is Mij, then the cofactor of element would be: C_ij = ((-1)_i+j)M_ij

---

Determinants

[Click Here for Sample Questions]

The matrix is a scalar (number) that is obtained from the matrix element by specified operations, which is characteristic of the matrix. This can be used only for square matrices.

The determinant of the (2 x 2) matrix:

A = \(\begin{bmatrix}a_{11} & a_{12} \\[0.3em]a_{21} & a_{22} \\[0.3em] \end{bmatrix}\)

is given by det A 

|A| = \(\begin{bmatrix}a_{11} & a_{12} \\[0.3em]a_{21} & a_{22} \\[0.3em] \end{bmatrix}\)

= a₁₁ * a₂₂ – a₁₂ * a₂₁

Read More: Identity Matrix

---

Different Types of Matrices

[Click Here for Previous Year Questions]

There are different types of matrices mentioned below:

1. Row matrix – These are the matrices that possess one row. A = [a_ij]_1×n
2. Column matrix – These are the matrix which possesses m rows and one column. A = [a_ij]_m×1
3. Null/Zero matrix – The matrix where all components are zero is termed as zero matrix. A = [a_ij]_mxn where, a_ij = 0
4. Singleton Matrix – Any matrix where it has only one element are termed as Singleton matrix A = [a_ij]_mxn where, m = n =1
5. Horizontal matrix – A matrix is termed as horizontal matrix when the number of columns is greater than the number of rows [a_ij]_mxn where n > m
6. Vertical matrix - A matrix is termed as vertical matrix, if the number of rows is greater than the number of columns [a_ij]_mxn where, m > n
7. Square matrix – The number of rows is equivalent to the number of columns [a_ij]_mxn where, m = n
8. Diagonal Matrix – In case of square matrix in which all the components are zero excluding those in the principal diagonal where A = [a_ij] when i ≠ j
9. Rectangular Diagonal Matrix – It is a type of a matrix that has one leading diagonal with numbers and the rest of the records are zeros.
10. Scalar Matrix – In case of a diagonal matrix where all the principal diagonal components are equal is known as scalar matrix where A = [a_ij]_mxn where,a_ij = { \( \begin{matrix} 0, & i \neq j \\ k, & i = j \\ \end{matrix}\) }
11. Identity Matrix or Unit Matrix - In case of a diagonal matrix where A = [a_ij]_mxn where,{\( \begin{matrix} 0, & i \neq j \\ k, & i = j \\ \end{matrix}\)}

Read More: Transpose of a Matrix

---

Minors and Cofactors

[Click Here for Sample Questions]

A cofactor is a number that will remove the column and row of a value in a matrix. It is essential to understand minors and cofactor matrices. It is essential to understand minors and cofactor matrices so that you can solve complex problems relating to determinants.

---

Cofactor Formula

[Click Here for Previous Year Questions]

The formula used any matrix of order n x n and M_ij be the (n – 1) x (n – 1) matrix obtained by deleting the ith row and jth column. The co-factor of C_ij of a_ij can be found using the formula:

C_ij = ((-1)^i+j) det (M_ij)

The cofactor is always represented with +ve (positive) or -ve (negative) signs. 

---

Solved Questions

[Click Here for Sample Questions]

Ques: If Δ = \(\begin{bmatrix}4 & 7 & 2 \\[0.3em]9 & 5 & 8 \\[0.3em]1 & 2 & 5 \\[0.3em] \end{bmatrix}\)in a determinant, then write the minor of the element a₂₃? (1 mark)

Ans: Minor of the element a₂₃= 8 – 7 = 1

Ques : Find the minors and cofactors of all the elements of the determinant  \(\begin{bmatrix}6 & 3 \\[0.3em]4 & 8 \\[0.3em]\end{bmatrix}\)? (3 marks)

Ans: Minor of the element a_ij is M_ij.

Here a₁₁ = 6. So M₁₁ = Minor of a₁₁ = 8

M₁₂ = Minor of the element a₁₂ = 4

M₂₁ = Minor of the element a₂₁ = 3

M₂₂ = Minor of the element a₂₂ = 6

Using the formula of cofactor of element would be: C_ij = ((-1)^i+j)M_ij

Now, the cofactor of a_ij is A_ij. So,

A₁₁ = (–1)¹⁺¹, M₁₁ = (–1)² (8) = 8

A₁₂ = (–1)¹⁺², M₁₂ = (–1)³ (4) = – 4

A₂₁ = (–1)²⁺¹, M₂₁ = (–1)³ (3) = – 3

A₂₂ = (–1)²⁺², M₂₂ = (–1)⁴ (6) = 6

Ques 3: If Δ = \(\begin{bmatrix}3 & 5 & 6 \\[0.3em]6 & 2 & 9 \\[0.3em]4 & 8 & 1 \\[0.3em] \end{bmatrix}\) , write:

(i) the minor of the element a₂₃

(ii) the cofactor of the element a³² (3 marks)

Ans 3:  (i) minor of the element M₂₃ = 24 – 20 = 4

(ii) Using the formula of cofactor of element would be: C_ij = ((-1)^i+j)M_ij

the cofactor of the element a₃₂ = A₂₃ = (–1)²⁺³, M₁₁ = (–1)⁵ (4) = - 4

Read More:

---

Things to Remember

• Cofactor of an element of a matrix is the determinant of the matrix 
• It is obtained by multiplying the corner of diagonals in a matrix 
• Eliminate the adjacent row and columns of the element placed at the centre position of the matrix and then multiplying by +1 or -1 according to the position of the element.
• To find the adjoint of the matrix and the inverse of the given matrix the cofactor of a matrix is used.
• Cofactor is always represented with +ve (positive) or -ve (negative) signs.
• There are six types of matrices such as row matrix, Column matrix, Null/Zero matrix, Horizontal matrix, vertical matrix and square matrix.
• The cofactor formula where any matrix of order n x n and Mij be the (n – 1) x (n – 1) matrix.

C_ij = ((-1)^i+j) det (M_ij)

---

Previous Year Questions

1. Extraction of metal from the ore cassiterite involves...[JEE Advanced 2011]
2. Commonly used vectors for human genome sequencing are...[NEET UG 2014]
3. Interfascicular cambium and cork cambium are formed due to​..
4. Pneumotaxic centre is present in​...[UP CPMT 2007]
5. Reaction of HBr with propene in the presence of peroxide gives….[NEET UG 2004]
6. Assuming the expression for the pressure exerted by the gas on the walls of the container, it can be shown that pressure is...[MHT CET 2016]
7. Which among the following is the strongest acid?...[TS EAMCET 2017]
8. Isopropyl alcohol on oxidation forms​..
9. A vector is not changed if​..
10. Which of the following arrangements does not represent the correct order of the property stated against it?...[JEE Main 2013]
11. The major product of the following reaction is​...[JEE Main 2019]
12. Major product of the following reaction is..[JEE Main 2023]
13. The percentage of nitrogen in urea is about..
14. The electric field at a point is​
15. Which of the following statements is true?​..[JKCET 2006]