Sample Problems on Cofactors of a Matrix

Problem 1: If a matrix A is  [Tex]\bold{|A| = \begin{vmatrix} 5&3&1\\ 2&0&-1\\1&2&3 \end{vmatrix}}    [/Tex]write the cofactor of the element a_32.

Solution:

As asked in question we have to find the co factor of element a32 which means our row (i) = 3 and column (j) = 2 so we have row and column as we do to find the minor by deleting the rows and column at which asked element exist we do the same in this question to and then put that in our formula -> A_ij = (-1)^i+j  M_ij

Cofactor of elements [Tex]= (-1)^{3+2}\begin{vmatrix}5 & 1\\2&-1 \end{vmatrix}[/Tex]

⇒ Cofactor of element = (-1)(-5 – 2) = 7

So after putting in the formula of finding cofactor and doing expansion of determinant we get (-1) (-5 – 2) which on solving gives the answer 7, this is our required answer.

Problem 2: If A_ij of the element a_ij of the determinant is given below, then write the value of a₃₂ . A₃₂.

[Tex] \bold{|A| = \begin{vmatrix} 2&-3&5\\ 6&0&4\\1&5&-7 \end{vmatrix}} [/Tex]

Solution: 

In the question, we are having determinant. So we have row and column given in the question.

Here, a₃₂ = 5 (second element of third row)

Given, A_ij is the cofactor of the element a_ij of A . So now we can solve this question by putting the values in the formula of cofactor as discussed in above question.

Cofactor of elements (A₃₂)[Tex]= (-1)^{3+2}\begin{vmatrix}2 &5\\6&4 \end{vmatrix} [/Tex]

⇒ Cofactor of element (A₃₂) = (-1)(8 – 30) = 22

 a₃₂ . A₃₂ = 5 . 22 = 110

So, 110 is our required answer.

Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)^i+j times the minor of the matrix.

They are the backbones of Linear Algebra and are used to find the value of a matrix’s determinant, adjoint, and inverse. Other than that there are many use cases in computer science for minors and cofactors. In this article, we will study minors and cofactors in detail. Other than that, we will also learn about the determinants, matrix inversion, and many more.