How do you calculate the minors and cofactors of a 3x3 matrix? Provide an example.
|Mathematics
How do you calculate the minors and cofactors of a 3x3 matrix? Provide an example.
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Answer
MA=−3−6−3−6−12−6−3−6−3
Hey Gezahagn, good to see you again.
To find the minors and cofactors of a matrix, we use the following definitions:
• The minorMij of an element aij is the determinant of the submatrix formed by deleting the i-th row and j-th column.
• The cofactorCij of an element aij is given by Cij=(−1)i+jMij.
Let's calculate them for both matrices.
a) For matrix A=147258369:
Step 1: Calculate the minors Mij.
M11=det(5869)=(5)(9)−(6)(8)=45−48=−3M12=det(4769)=(4)(9)−(6)(7)=36−42=−6M13=det(4758)=(4)(8)−(5)(7)=32−35=−3M21=det(2839)=(2)(9)−(3)(8)=18−24=−6M22=det(1739)=(1)(9)−(3)(7)=9−21=−12M23=det(1728)=(1)(8)−(2)(7)=8−14=−6M31=det(2536)=(2)(6)−(3)(5)=12−15=−3M32=det(1436)=(1)(6)−(3)(4)=6−12=−6M33=det(1425)=(1)(5)−(2)(4)=5−8=−3
The matrix of minors for A is:
MA=−3−6−3−6−12−6−3−6−3
Step 2: Calculate the cofactors Cij=(−1)i+jMij.
C11=(−1)1+1(−3)=−3C12=(−1)1+2(−6)=6C13=(−1)1+3(−3)=−3C21=(−1)2+1(−6)=6C22=(−1)2+2(−12)=−12C23=(−1)2+3(−6)=6C31=(−1)3+1(−3)=−3C32=(−1)3+2(−6)=6C33=(−1)3+3(−3)=−3
The matrix of cofactors for A is:
CA=−36−36−126−36−3
b) For matrix B=100130345:
Step 1: Calculate the minors Mij.
M11=det(3045)=(3)(5)−(4)(0)=15−0=15M12=det(0045)=(0)(5)−(4)(0)=0−0=0M13=det(0030)=(0)(0)−(3)(0)=0−0=0M21=det(1035)=(1)(5)−(3)(0)=5−0=5M22=det(1035)=(1)(5)−(3)(0)=5−0=5M23=det(1010)=(1)(0)−(1)(0)=0−0=0M31=det(1334)=(1)(4)−(3)(3)=4−9=−5M32=det(1034)=(1)(4)−(3)(0)=4−0=4M33=det(1013)=(1)(3)−(1)(0)=3−0=3
The matrix of minors for B is:
MB=155−5054003
Step 2: Calculate the cofactors Cij=(−1)i+jMij.
C11=(−1)1+1(15)=15C12=(−1)1+2(0)=0C13=(−1)1+3(0)=0C21=(−1)2+1(5)=−5C22=(−1)2+2(5)=5C23=(−1)2+3(0)=0C31=(−1)3+1(−5)=−5C32=(−1)3+2(4)=−4C33=(−1)3+3(3)=3
The matrix of cofactors for B is:
CB=15−5−505−4003
What's next? Send 'em!
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Hey Gezahagn, good to see you again. To find the minors and cofactors of a matrix, we use the following definitions: • The minor M_ij of an element a_ij is the determinant of the submatrix formed by deleting the i-th row and j-th column.
How do you calculate the minors and cofactors of a 3x3 matrix? Provide an example.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
