Minor and Cofactors of a Matrix

In linear algebra, the Cofactors of a matrix is an arrangement of numbers, symbols or expressions called entries of the matrix, in the form of m rows and n columns. Matrices are rectangular array representations of elements which are used to represent data in two-dimensional form.

There are some special types of matrices:

• Square matrix: A matrix with the same number of rows and columns.
•  Row matrix: Matrix of order 1 × n, which has only one row and n columns.
• Column matrix: Matrix of order m × 1, that is, it has only one column and m rows.
• Identity matrix: An identity matrix I_n  is a square matrix of order n whose diagonal elements are 1 and the rest all the entries are zero.
• Null matrix or zero matrix: A matrix whose all elements are zero.

In this article, we will discuss invertible matrices – conditions for the invertibility of a matrix and how to find the inverse of a matrix.

Invertible Matrices

A non-zero square matrix A is said to be invertible if and only if there exists a square matrix B such that AB = BA = I, where I is an identity matrix of the same order as that of A and B.

Then we write the inverse of A as A^–1 = B.

Condition for invertibility

The condition for invertibility of a matrix is that the square matrix should be non-singular. That is, the determinant of the given matrix should not be zero.

Finding Inverse Using Cofactors

Cofactors of an element a_ij (ith row and jth column) of a matrix is a number obtained by taking the determinant of the elements by eliminating the entries of the row and column which contain the element a_ij.

Learn more about how to find cofactors of a given matrix here.  

The sub-matrix obtained by eliminating row and column containing the element a_ij is called minor M_ij.

Thus, we get the formula for the cofactors of an n × n matrix as:

C_ij = ( –1)^{i + j}det(M_ij) where i, j = 1, 2, 3, …, n.

By evaluating all the cofactors corresponding to each matrix element, we find the adjoint of the given matrix, which is the transpose of the matrix whose elements are cofactors of each element of the original matrix.

Hence, the inverse of matrix A is given by

A^–1 = 1/|A| (adj A), where |A| is determinant of A.

Thus, the steps to find inverse using cofactors:

• Find cofactors corresponding to each element of the given matrix using the formula

C_ij = ( –1)^{i + j}det(M_ij) where i, j = 1, 2, 3, …, n for element a_ij,

• Form another matrix with cofactors of each element.
• Take the transpose of this matrix to find the adjoint of A.
• Finally, use the following formula to find the inverse:

A^–1 = 1/|A| (adj A), where |A| ≠ 0 is the determinant of A.

Note: We must not confuse cofactors with factors. Factors of a number are such numbers which are multiplied together to get the given number. For example, factors of 6 are 2 and 3, 6 = 2 × 3.  Cofactors are numbers associated with each element of a matrix.

Learn more about the factor of a number here.

Finding Inverse Using Elementary Operations

Another method of finding the inverse is by using elementary operations. Following are elementary operations:

• Any two rows (columns) of a matrix are interchangeable.
• Rows (columns) of a matrix can be multiplied by a non-zero scalar.
• Addition of two rows (columns) by multiplication of each element of row (column) by a non-zero scalar.

By applying all these elementary operations, we can obtain an equivalent matrix to the original matrix.

Steps to find inverse by elementary row transformation of a square matrix A of order n × n:

• Write A = I. A where I is a unit matrix of order the same as that of A.
• Perform elementary row operations on both sides to transform A into a unit matrix while keeping A on R.H.S constant.
• Consequently, it will transform into matrix B, which is the inverse of A. 

Therefore, A = I. A

After performing elementary row transformations,

I = B. A ⇒ B = A^–1.

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