Minors and Cofactors of a Matrix

Minors and Cofactors of a Matrix

Minors and Cofactors of a Matrix

Minors:

Let A be a square matrix and aij is the element in ith row and jth column of A. Then the minor of the element aij is the determinant of the matrix formed by omitting ith row and jth column of A. The minor of element aij is denoted by Mij.


Minors For 2×2 Matrix

Let A = (■(a_11&a_12@a_21&a_22 )) be a 2×2 matrix. Then the minor of the element a11 is the determinant of matrix formed by omitting 1st row and 1st column of A. i.e. M11 = |a22|. Similarly, minor of a12 is M12 =|a21| and so on.

Look at the following pattern to find the minors of the elements of a 2×2 matrix:

Pattern to find the minors of the elements of a 2×2 matrix.

Thus,

M11 = minor of a11 = |a22| = a22

M12 = minor of a12 = |a21| = a21

M21 = minor of a21 = |a12| = a12

M22 = minor of a22 = |a11| = a11

 

Minors For 3×3 Matrix                                                          

Let A = (■(a_11&a_12&a_13@a_21&a_22&a_23@a_31&a_32&a_33 )) be a 3×3 matrix. Then the minor of the element a11 is the determinant of the matrix formed by omitting 1st row and 1st column of A. i.e. M11 = |■(a_22&a_23@a_32&a_33 )|. Similarly, minor of a12 is M12 = |■(a_21&a_23@a_31&a_33 )| and so on.

Look at the following pattern to find the minors of the elements of a 3×3 matrix:

Pattern to find the minors of the elements of a 3×3 matrix.

Thus,

M11 = minor of a11 = |■(a_22&a_23@a_32&a_33 )| = a22a33 – a23a32 M12 = minor of a12 = |■(a_21&a_23@a_31&a_33 )| = a21a33 – a23a31 M13 = minor of a13 = |■(a_21&a_22@a_31&a_32 )| = a21a32 – a22a31 M21 = minor of a21 = |■(a_12&a_13@a_32&a_33 )| = a12a33 – a13a32 M22 = minor of a22 = |■(a_11&a_13@a_31&a_33 )| = a11a33 – a13a31 M23 = minor of a23 = |■(a_11&a_12@a_31&a_32 )| = a11a32 – a12a31 M31 = minor of a31 = |■(a_12&a_13@a_22&a_23 )| = a12a23 – a13a22 M32 = minor of a32 = |■(a_11&a_13@a_21&a_23 )| = a11a23 – a13a21 M33 = minor of a33 = |■(a_11&a_12@a_21&a_22 )| = a11a22 – a12a21


Cofactors:

Let A be a square matrix and aij is the element in ith row and jth column of A. Then the cofactor of the element aij is given by the number (-1)i+j Mij, where Mij is the minor of the element aij. The cofactor of element aij is denoted by Aij.


Cofactors For 2×2 Matrix

Let A = (■(a_11&a_12@a_21&a_22 )) be a 2×2 matrix. Thus, the cofactors of the elements of A are as follows: A11 = cofactor of a11 = (-1)1+1 M11 = |a22| = a22 A12 = cofactor of a12 = (-1)1+2 M12 = -|a21| = - a21 A21 = cofactor of a21 = (-1)2+1 M21 = -|a12| = - a12 A22 = cofactor of a22 = (-1)2+2 M22 = |a11| = a11


Cofactors For 3×3 Matrix                                                  

Let A = (■(a_11&a_12&a_13@a_21&a_22&a_23@a_31&a_32&a_33 )) be a 3×3 matrix. Thus, the cofactors of the elements of A are as follows: A11 = cofactor of a11 = (-1)1+1 M11 = |■(a_22&a_23@a_32&a_33 )| A12 = cofactor of a12 = (-1)1+2 M12 = – |■(a_21&a_23@a_31&a_33 )| A13 = cofactor of a13 = (-1)1+3 M13 = |■(a_21&a_22@a_31&a_32 )| A21 = cofactor of a21 = (-1)2+1 M21 = – |■(a_12&a_13@a_32&a_33 )| A22 = cofactor of a22 = (-1)2+2 M22 = |■(a_11&a_13@a_31&a_33 )| A23 = cofactor of a23 = (-1)2+3 M23 = – |■(a_11&a_12@a_31&a_32 )| A31 = cofactor of a31 = (-1)3+1 M31 = |■(a_12&a_13@a_22&a_23 )| A32 = cofactor of a32 = (-1)3+2 M32 = – |■(a_11&a_13@a_21&a_23 )| A33 = cofactor of a33 = (-1)3+3 M33 = |■(a_11&a_12@a_21&a_22 )|


Worked Out Examples

Example 1: Find the minor and cofactor of matrix (■(1&2@3&4)). Solution: Here, Let A = (■(1&2@3&4)) = (■(a_11&a_12@a_21&a_22 ))  Then,  a11 = 1, a12 = 2, a21 = 3 and a22 = 4 Thus, M11 = minor of a11 = |4| = 4 M12 = minor of a12 = |3| = 3 M21 = minor of a21 = |2| = 2 M22 = minor of a22 = |1| = 1 And, A11 = cofactor of a11 = (-1)1+1 M11 = 4 A12 = cofactor of a12 = (-1)1+2 M12 = -3 A21 = cofactor of a21 = (-1)2+1 M21 = -2 A22 = cofactor of a22 = (-1)2+2 M22 = 1


Example 2: Find the minors and cofactors of matrix (■(1&2&-3@2&0&4@3&2&1)). Solution: Here, Let A = (■(1&2&-3@2&0&4@3&2&1)) = (■(a_11&a_12&a_13@a_21&a_22&a_23@a_31&a_32&a_33 )) Then, a11 = 1		a12 = 2		a13 = -3 a21 = 2		a22 = 0		a23 = 4 a31 = 3		a32 = 2		a33 = 1 Thus, M11 = minor of a11 = |■(0&4@2&1)| = 0 – 8 = - 8 M12 = minor of a12 = |■(2&4@3&1)| = 2 – 12 = - 10 M13 = minor of a13 = |■(2&0@3&2)| = 4 – 0 = 4 M21 = minor of a21 = |■(2&-2@2&1)| = 2 + 4 = 8 M22 = minor of a22 = |■(1&-3@3&1)| = 1 + 9 = 10 M23 = minor of a23 = |■(1&2@3&2)| = 2 – 6 = - 4 M31 = minor of a31 = |■(2&-3@0&4)| = 8 – 0 = 8 M32 = minor of a32 = |■(1&-3@2&4)| = 4 + 6 = 10 M33 = minor of a33 = |■(1&2@2&0)| = 0 – 4 = - 4 And, A11 = cofactor of a11 = (-1)1+1 M11 = - 8 A12 = cofactor of a12 = (-1)1+2 M12 = - (- 10) = 10 A13 = cofactor of a13 = (-1)1+3 M13 = 4 A21 = cofactor of a21 = (-1)2+1 M21 = - 8 A22 = cofactor of a22 = (-1)2+2 M22 = 10 A23 = cofactor of a23 = (-1)2+3 M23 = - (- 4) = 4 A31 = cofactor of a31 = (-1)3+1 M31 = 8 A32 = cofactor of a32 = (-1)3+2 M32 = - 10 A33 = cofactor of a33 = (-1)3+3 M33 = - 4

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