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Question 1196138: Given P = ( 3 0 2
2 1 2
-3 -4 -6 )
a) if ( 2 6 n
-2 m 9
0 -2 1 ) is the cofactor of matrix P, find the values of m and without finding a new cofactor matrix.
b) find the adjoint matrix of P hence, find P-1 (identity P)
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **a) Finding the values of m and n**
* **Understand Cofactor Matrix:**
* The cofactor of an element in a matrix is calculated by:
* Finding the minor (determinant of the submatrix obtained by removing the row and column of the element).
* Multiplying the minor by (-1)^(i+j), where i and j are the row and column indices of the element.
* **Determine Cofactors:**
* We are given the following cofactors:
* Cofactor of the element '3' (row 1, column 1) = 2
* Cofactor of the element '0' (row 1, column 2) = 6
* Cofactor of the element '2' (row 1, column 3) = n
* Cofactor of the element '2' (row 2, column 1) = -2
* Cofactor of the element '1' (row 2, column 2) = m
* Cofactor of the element '2' (row 2, column 3) = 9
* Cofactor of the element '-3' (row 3, column 1) = 0
* Cofactor of the element '-4' (row 3, column 2) = -2
* Cofactor of the element '-6' (row 3, column 3) = 1
* **Calculate Cofactors:**
* **Cofactor of '3':**
* Minor: det([ 1 2 ; -4 -6 ]) = 1*(-6) - 2*(-4) = -6 + 8 = 2
* Cofactor: (-1)^(1+1) * 2 = 1 * 2 = 2 (Matches the given value)
* **Cofactor of '0':**
* Minor: det([ 2 2 ; -1 -6 ]) = 2*(-6) - 2*(-1) = -12 + 2 = -10
* Cofactor: (-1)^(1+2) * (-10) = -1 * (-10) = 10
* **Therefore, n = 10**
* **Cofactor of '2':**
* Minor: det([ 2 1 ; -1 -4 ]) = 2*(-4) - 1*(-1) = -8 + 1 = -7
* Cofactor: (-1)^(1+3) * (-7) = 1 * (-7) = -7
* **Therefore, n = -7**
* **Cofactor of '2':**
* Minor: det([ 0 2 ; -4 -6 ]) = 0*(-6) - 2*(-4) = 8
* Cofactor: (-1)^(2+1) * 8 = -1 * 8 = -8
* **Therefore, m = -8**
**b) Find the Adjoint Matrix of P and P-1**
* **Find the Cofactor Matrix:**
* Using the calculated cofactors and the remaining cofactors (which you can calculate similarly), construct the cofactor matrix:
[ 2 6 -7
-2 -8 9
0 -2 1 ]
* **Find the Adjoint Matrix:**
* The adjoint of a matrix is the transpose of its cofactor matrix.
[ 2 -2 0
6 -8 -2
-7 9 1 ]
* **Find the Determinant of P:**
* det(P) = 3 * det([ 1 2 ; -4 -6 ]) - 0 * det([ 2 2 ; -1 -6 ]) + 2 * det([ 2 1 ; -1 -4 ])
* det(P) = 3 * 2 - 0 * (-10) + 2 * (-7)
* det(P) = 6 - 14
* det(P) = -8
* **Find the Inverse of P (P-1):**
* P-1 = (1/det(P)) * adj(P)
* P-1 = (-1/8) * [ 2 -2 0
6 -8 -2
-7 9 1 ]
* P-1 = [ -1/4 1/4 0
-3/4 1 1/4
7/8 -9/8 -1/8 ]
**Therefore:**
* **m = -8**
* **n = -7**
* **The adjoint matrix of P is:**
[ 2 -2 0
6 -8 -2
-7 9 1 ]
* **The inverse of P (P-1) is:**
[ -1/4 1/4 0
-3/4 1 1/4
7/8 -9/8 -1/8 ]
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