Minors and cofactors are two of the most important concepts in matrices as they are crucial in finding the adjoint and the inverse of a matrix. To find the determinants of a large square matrix (like 4×4), it is important to find the minors of that matrix and then the cofactors of that matrix. Below is a detailed explanation on “what are minors and cofactors” along with steps to find them.
Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. e.g. in the determinant D=∣∣∣∣∣∣∣a11a21a31a12a22a32a13a23a33∣∣∣∣∣∣∣,
minor of a12 is denoted as M12=∣∣∣∣∣a21a31a23a33∣∣∣∣∣
and so on.
What are Cofactors?
Cofactor of an element aij is related to its minor as Cij=(−1)i+jMij, where ‘i’ denotes the ith row and ‘j’ denotes the jth column to which the element aij belongs.
Now we define the value of the determinant of order three in terms of ‘Minor’ and ‘Cofactor’ as
Practice Problems on How to Find Minors and Cofactors
Question 1: Find the cofactor of a12 in the following ∣∣∣∣∣∣∣261−30554−7∣∣∣∣∣∣∣
In this problem we have to find the cofactor of a12 therefore eliminate all the elements of the first row and the second column and by obtaining the determinant of remaining elements we can calculate the cofactor of a12
Here a12= Element of first row and second column = –3
M12= Minor of a12 = ∣∣∣∣∣614−7∣∣∣∣∣= 6 (-7) – 4(1)
= -42 – 4 = -46
Cofactor of (−3)=(−1)1+2(−46)=−(−46)=46
Question 2: Write the minors and cofactors of the elements of the following determinants:
By eliminating row and column of an element, the remaining is the minor of the element.
Cofactor of 2=−11+1M11=+3M12=Minorofelement(−4)=∣∣∣∣∣∣∣∣2…−4⋮03∣∣∣∣∣∣∣∣=0;Cofactorof(−4)=(−1)1+2M12=(−1)0=0M21=Minorofelement(0)=∣∣∣∣∣∣∣∣2−4⋮0…3∣∣∣∣∣∣∣∣=−4;Cofactorof(0)=(−1)2+1M21=(−1)(−4)=4M22=Minorofelement(3)=∣∣∣∣∣∣∣∣2−4⋮0…3∣∣∣∣∣∣∣∣=2;Cofactorof(3)=(−1)2+2M22=+2
Question 3: Find the minor and cofactor of each element of the determinant ∣∣∣∣∣∣∣212−24135−3∣∣∣∣∣∣∣
By eliminating the row and column of an element, the determinant of remaining elements is the minor of the element, i.e. Mi×j and by using formula (−1)i+jMi×j we will get the cofactor of the element.
The minors are M11=∣∣∣∣∣415−3∣∣∣∣∣=−17,M12=∣∣∣∣∣125−3∣∣∣∣∣=−13,M13=∣∣∣∣∣1241∣∣∣∣∣=−7M21=∣∣∣∣∣−213−3∣∣∣∣∣=3,M22=∣∣∣∣∣223−3∣∣∣∣∣=−12,M23=∣∣∣∣∣22−21∣∣∣∣∣=6,M31=∣∣∣∣∣−2435∣∣∣∣∣=−22,M32=∣∣∣∣∣2135∣∣∣∣∣=7,M33=∣∣∣∣∣21−24∣∣∣∣∣=10