Minors and Cofactor: Overview, Questions, Preparation

Minor of an element is the determinant of the matrix obtained by deleting a specific row and column from a larger matrix.

If A = $\left[a_{ij}\right]$  (square matrix)

Minor $M_{ij}$ = determinant obtained by deleting i^th row and j^th column of matrix A.

Similarly, cofactor is a term similar to that of a minor. In simple terms, it is a minor multiplied by (–1) ^i+j which gives a sign (positive or negative) to the element.

The cofactors $A_{ij}$ of an element $a_{ij}$ of a square matrix A = $\left[a_{ij}\right]$ can be defined as,

$A_{ij}$ = (–1) ^{i+j      $M_{ij}$}

Let us understand the concept of minors and cofactors through the following cases.

Case 1: A 2x2 Matrix

Consider the matrix A = $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$

Then,

Minor of $a_{11}$ = $M_{11}$ = 4               Cofactors of $a_{11}$ = $A_{11}$ = (–1) ¹⁺¹ $M_{11}$ = 4

Minor of $a_{12}$ = $M_{12}$ = 3               Cofactors of $a_{12}$ = $A_{12}$ = (–1) ¹⁺² $M_{12}$ = – 3

Minor of $a_{21}$ = $M_{21}$ =2                Cofactors of $a_{21}$ = $A_{21}$ = (–1) ²⁺¹ $M_{21}$ = – 2

Minor of $a_{22}$ = $M_{22}$ = 1              Cofactors of $a_{22}$ = $A_{22}$ = (–1) ²⁺² $M_{22}$ = 1

Case 2: A 3x3 Matrix $b_{23}$

Now, consider the matrix B = $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right]$ . Then,

Minor of $b_{11}$ = $M_{11}$ = $\left|\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 9\hfill \end{array}\right|$ = – 3           Cofactors of $b_{11}$ = $B_{11}$ = (–1) ¹⁺¹ $M_{11}$ = – 3

Minor of $b_{12}$ = $M_{12}$ = $\left|\begin{array}{cc}\hfill 4\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 9\hfill \end{array}\right|$ = – 6           Cofactors of $b_{12}$ = $B_{12}$ = (–1) ¹⁺² $M_{12}$ = 6

Minor of $b_{13}$ = $M_{13}$ = $\left|\begin{array}{cc}\hfill 4\hfill & \hfill 5\hfill \\ \hfill 7\hfill & \hfill 8\hfill \end{array}\right|$ = – 3            Cofactors of $b_{13}$ = $B_{13}$ = (–1) ¹⁺³ $M_{13}$ = – 3

Minor of $b_{21}$ = $M_{21}$ = $\left|\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill 8\hfill & \hfill 9\hfill \end{array}\right|$ = – 6            Cofactors of $b_{21}$ = $B_{21}$ = (–1) ²⁺¹ $M_{21}$ = 6

Minor of $b_{22}$ = $M_{22}$ = $\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 7\hfill & \hfill 9\hfill \end{array}\right|$ = –12           Cofactors of $b_{22}$ = $B_{22}$ = (–1) ²⁺² $M_{22}$ = – 12

Minor of $b_{23}$ = $M_{23}$ = $\left|\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 7\hfill & \hfill 8\hfill \end{array}\right|$ = – 6          Cofactors of $b_{23}$ = $B_{23}$ = (–1) ²⁺³ $M_{23}$ = 6

Minor of $b_{31}$ = $M_{31}$ = $\left|\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill 5\hfill & \hfill 6\hfill \end{array}\right|$ = – 3           Cofactors of $b_{31}$ = $B_{31}$ = (–1) ³⁺¹ $M_{31}$ = – 3

Minor of $b_{32}$ = $M_{32}$ = $\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{array}\right|$ = – 6          Cofactors of $b_{32}$ = $B_{32}$ = (–1) ³⁺² $M_{32}$ = 6

Minor of $b_{33}$ = $M_{33}$ = $\left|\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 5\hfill \end{array}\right|$ = – 3           Cofactors of $b_{33}$ = $B_{33}$ = (–1) ³⁺³ $M_{33}$ = –3

Finding the minor of an element involves some simple steps. To do so, you need to eliminate the row and column containing that specific element. First, choose the element for which you wish to find the minor. For example, if you have a 3x3 matrix and want to find the minor of the element in the first row first column, eliminate the first row and first column.

After that, you will get a smaller matrix comprising of the remaining elements. Simply calculate the determinant of the remaining 2x2 matrix. 

Cofactor is nothing but simply a signed minor that is determined by the position of the element in the original matrix. Simply calculate the minor of the element first in order to proceed with the cofactor steps.

After finding the minor, put the value into the cofactor formula and look for the following point in the end-result:

• If (i+j) is even, the cofactor is the same as the minor. 
• If (i+j) is odd, the cofactor is the negative of the minor. 

Solve the following problem to clear your basic concepts related to minors and cofactors.

Question: Write the minors and cofactors of the elements of $\left[\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 4\hfill & \hfill 3\hfill \end{array}\right]$ .

Answer: Let A = $\left[a_{ij}\right]$ = $\left[\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 4\hfill & \hfill 3\hfill \end{array}\right]$ . Then,

Minor of $a_{11}$ = $M_{11}$ = 3               Cofactors of $a_{11}$ = $A_{11}$ = (–1) ¹⁺¹ $M_{11}$ = 3

Minor of $a_{12}$ = $M_{12}$ = 4               Cofactors of $a_{12}$ = $A_{12}$ = (–1) ¹⁺² $M_{12}$ = – 4

Minor of $a_{21}$ = $M_{21}$ = – 2           Cofactors of $a_{21}$ = $A_{21}$ = (–1) ²⁺¹ $M_{21}$ = 2

Minor of $a_{22}$ = $M_{22}$ = 1              Cofactors of $a_{22}$ = $A_{22}$ = (–1) ²⁺² $M_{22}$ = 1

Minors and Cofactors are an important part of the chapter determinants and have the following mentioned applications in the subject:

1.  Inverse of a Matrix: Finding the inverse of a matrix involves calculating the adjoint matrix i.e. transpose of the cofactor matrix. 
2.  Linear Equations: An important principle called Cramer's rule relies on determinants, which are calculated using minors and cofactors. Cramer’s rule is used to solve systems of linear equations.
3.  Computer Science: Minors and cofactors have numerous applications in major areas of computer science, such as in algorithms, discrete mathematics, machine learning, and data structures involving matrices. 
4.  Engineering: Scientists and engineers also use minors and cofactors to solve some of the complex equations which are used in designing and manufacturing of equipments.