HOW TO FIND MINORS AND COFACTORS OF A MATRIX

Example 1 :

Find the minor and cofactor of the following matrix

Solution :

Minor of a₁₁ (Ignore 1^st row and 1^st column)

Minor of a₁₁  =  -6+4  ==>  -2

Minor of a₁₂  =  0-10  ==>  -10

Minor of a₁₃  =  0+5  ==>  5

Minor of a₂₁  =  24+2  ==>  26

Minor of a₂₂  =  18-5  ==>  13

Minor of a₂₃  =  -6-20  ==>  -26

Minor of a₃₁  =  8+1  ==>  9

Minor of a₃₂  =  6-0  ==>  6

Minor of a₃₃  =  -3-0  ==>  -3 

Example 2 :

Find the minor and cofactor of the following matrix

Solution :

Minor of a₁₁  =  2-0  ==>  2

Minor of a₁₂  =  -2-0  ==>  -2

Minor of a₁₃  =  4+2  ==>  6

Minor of a₂₁  =  -1+2  ==>  1

Minor of a₂₂  =  -3+1  ==>  -2

Minor of a₂₃  =  6-1  ==>  5

Minor of a₃₁  =  0-2  ==>  -2

Minor of a₃₂  =  0+2  ==>  2

Minor of a₃₃  =  -6-2  ==>  -8 

Example 3 :

Find the minors and cofactors of the elements of the second row in the determinant

Solution :

The elements in the second row will be a₂₁, a₂₂ and a₂₃

Minor of a_{21 :}

By leaving the second row and first column, then doing the cross multiplication.

= 48 - 0

= 48

Minor of a_{22 :}

By leaving the second row and second column, then doing the cross multiplication.

= 8 - 21

= -13

Minor of a₂₃ :

By leaving the second row and third column, then doing the cross multiplication.

= 0 - 42

= -42

Finding cofactors of second row :

Cofactor of a₂₁ = - minor of a₂₁

= -48

Cofactor of a₂₂ = + minor of a₂₂

= + (-13)

= -13

Cofactor of a₂₃ = - minor of a₂₃

= -(-42)

= 42

Example 4 :

Find the minors and cofactors of the elements of the second row of the determinant

Solution :

The elements in the second row will be a₂₁, a₂₂ and a₂₃

Minor of a_{21 :}

By leaving the second row and first column, then doing the cross multiplication.

= 18 - (-21)

= 18 + 21

= 39

Minor of a_{22 :}

By leaving the second row and second column, then doing the cross multiplication.

= 9 - 6

= 3

Minor of a₂₃ :

By leaving the second row and third column, then doing the cross multiplication.

= -7 - 4

= -11

Finding cofactors of second row :

Cofactor of a₂₁ = - minor of a₂₁

= -39

Cofactor of a₂₂ = + minor of a₂₂

= + 3

Cofactor of a₂₃ = - minor of a₂₃

= -(11)

= -11

Example 5 :

Evaluate each of the following determinants using cofactors:

Solution :

By expanding the first row with cofactors, we get

= +2[0 - (-8)] - 1[3 - 6] + 0[-4 - 0]

= 2[0+8] - 1[-3] + 0

= 16 + 3

= 19

So, the values of the determinant given is 19.

Example 6 :

Solution :

x[4 - 0] - 0[2 - 3] + 0[0 - 2] = 0

x(4) + 0 = 0

4x = 0

x = 0/4

x = 0

So, the value of x is 0.

Example 7 :

Solution :

x[9 - 3x] - 3[9 - 2x] + 3[9 - 6] = 0

x(9 - 3x) - 3(9 - 2x) + 3(3) = 0

9x - 3x² - 27 + 6x + 9 = 0

- 3x² + 6x + 9x + 9 - 27 = 0

- 3x² + 15x - 18 = 0

Dividing by -3, we get

x² - 5x + 6 = 0

(x - 3)(x - 2) = 0

x = 3 and x = 2

So, the values of x are 2 and 3.

Example 8 :

Solution :

x²[8 - 1] - x[0 - 3] + 1[0 - 6] = 28

x²[7] - x[- 3] + 1[- 6] = 28

7x² + 3x - 6 = 28

7x² + 3x - 6 - 28 = 0

7x² + 3x - 34 = 0

7x² - 14x + 17x - 34 = 0

7x(x - 2) + 17(x - 2) = 0

(7x + 17)(x - 2) = 0

 x = -17/7 and x = 2

So, the values of x are 2 and -17/7.

Example 9 :

Use the property to find another matrix whose determinant is equal to Δ.

Solution :

Expansion of first determinant :

-1[6 - 12] - 1[6 - 6] -3[12 - 12]

= -1(-6) - 1(0) - 3(0)

= 6

Expansion of second determinant :

2[12 - 12] - 1[0 - 0] -3[0 - 0]

= 2(0) - 1(0) - 3(0)

= 0

Expansion of third determinant :

2[6 - 6] + 1[0 - 0] -3[0 - 0]

= 2(0) - 1(0) - 3(0)

= 0

Expansion of fourth determinant :

2[12 - 12] + 1[0 - 0] + 1[0 - 0]

= 2(0) - 1(0) + 1(0)

= 0

Applying these values, we get

= 1(6) -4(0) + 8(0) -2(0)

= 6

So, the determinant is 6.

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