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How do you calculate the minors and cofactors of a 3x3 matrix? Provide an example.

Mathematics

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2 parts
a

MA=(−3−6−3−6−12−6−3−6−3)M_A = \begin{pmatrix} -3 & -6 & -3 \\ -6 & -12 & -6 \\ -3 & -6 & -3 \end{pmatrix}

CA=(−36−36−126−36−3)C_A = \begin{pmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{pmatrix}

b

MB=(1500550−543)M_B = \begin{pmatrix} 15 & 0 & 0 \\ 5 & 5 & 0 \\ -5 & 4 & 3 \end{pmatrix}

CB=(1500−550−5−43)C_B = \begin{pmatrix} 15 & 0 & 0 \\ -5 & 5 & 0 \\ -5 & -4 & 3 \end{pmatrix}

Step-by-step solution

4 steps

To find the minors and cofactors of a matrix, we use the following definitions:

  • The minor MijM_{ij} of an element aija_{ij} is the determinant of the submatrix formed by deleting the ii-th row and jj-th column.
  • The cofactor CijC_{ij} of an element aija_{ij} is given by Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}.

Let's calculate them for both matrices.

Part a· 2 steps

For matrix A=(123456789)A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

  1. Calculate the minors MijM_{ij}.

    M11=det⁡(5689)=(5)(9)−(6)(8)=45−48=−3M_{11} = \det \begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} = (5)(9) - (6)(8) = 45 - 48 = -3 M12=det⁡(4679)=(4)(9)−(6)(7)=36−42=−6M_{12} = \det \begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} = (4)(9) - (6)(7) = 36 - 42 = -6 M13=det⁡(4578)=(4)(8)−(5)(7)=32−35=−3M_{13} = \det \begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix} = (4)(8) - (5)(7) = 32 - 35 = -3 M21=det⁡(2389)=(2)(9)−(3)(8)=18−24=−6M_{21} = \det \begin{pmatrix} 2 & 3 \\ 8 & 9 \end{pmatrix} = (2)(9) - (3)(8) = 18 - 24 = -6 M22=det⁡(1379)=(1)(9)−(3)(7)=9−21=−12M_{22} = \det \begin{pmatrix} 1 & 3 \\ 7 & 9 \end{pmatrix} = (1)(9) - (3)(7) = 9 - 21 = -12 M23=det⁡(1278)=(1)(8)−(2)(7)=8−14=−6M_{23} = \det \begin{pmatrix} 1 & 2 \\ 7 & 8 \end{pmatrix} = (1)(8) - (2)(7) = 8 - 14 = -6 M31=det⁡(2356)=(2)(6)−(3)(5)=12−15=−3M_{31} = \det \begin{pmatrix} 2 & 3 \\ 5 & 6 \end{pmatrix} = (2)(6) - (3)(5) = 12 - 15 = -3 M32=det⁡(1346)=(1)(6)−(3)(4)=6−12=−6M_{32} = \det \begin{pmatrix} 1 & 3 \\ 4 & 6 \end{pmatrix} = (1)(6) - (3)(4) = 6 - 12 = -6 M33=det⁡(1245)=(1)(5)−(2)(4)=5−8=−3M_{33} = \det \begin{pmatrix} 1 & 2 \\ 4 & 5 \end{pmatrix} = (1)(5) - (2)(4) = 5 - 8 = -3

    The matrix of minors for A is:

    MA=(−3−6−3−6−12−6−3−6−3)\boxed{M_A = \begin{pmatrix} -3 & -6 & -3 \\ -6 & -12 & -6 \\ -3 & -6 & -3 \end{pmatrix}}

  2. Calculate the cofactors Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}.

    C11=(−1)1+1(−3)=−3C_{11} = (-1)^{1+1}(-3) = -3 C12=(−1)1+2(−6)=6C_{12} = (-1)^{1+2}(-6) = 6 C13=(−1)1+3(−3)=−3C_{13} = (-1)^{1+3}(-3) = -3 C21=(−1)2+1(−6)=6C_{21} = (-1)^{2+1}(-6) = 6 C22=(−1)2+2(−12)=−12C_{22} = (-1)^{2+2}(-12) = -12 C23=(−1)2+3(−6)=6C_{23} = (-1)^{2+3}(-6) = 6 C31=(−1)3+1(−3)=−3C_{31} = (-1)^{3+1}(-3) = -3 C32=(−1)3+2(−6)=6C_{32} = (-1)^{3+2}(-6) = 6 C33=(−1)3+3(−3)=−3C_{33} = (-1)^{3+3}(-3) = -3

    The matrix of cofactors for A is:

    CA=(−36−36−126−36−3)\boxed{C_A = \begin{pmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{pmatrix}}

Part b

MB=(1500550−543)M_B = \begin{pmatrix} 15 & 0 & 0 \\ 5 & 5 & 0 \\ -5 & 4 & 3 \end{pmatrix}

For matrix B=(113034005)B = \begin{pmatrix} 1 & 1 & 3 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{pmatrix}

  1. Calculate the minors MijM_{ij}.

    M11=det⁡(3405)=(3)(5)−(4)(0)=15−0=15M_{11} = \det \begin{pmatrix} 3 & 4 \\ 0 & 5 \end{pmatrix} = (3)(5) - (4)(0) = 15 - 0 = 15 M12=det⁡(0405)=(0)(5)−(4)(0)=0−0=0M_{12} = \det \begin{pmatrix} 0 & 4 \\ 0 & 5 \end{pmatrix} = (0)(5) - (4)(0) = 0 - 0 = 0 M13=det⁡(0300)=(0)(0)−(3)(0)=0−0=0M_{13} = \det \begin{pmatrix} 0 & 3 \\ 0 & 0 \end{pmatrix} = (0)(0) - (3)(0) = 0 - 0 = 0 M21=det⁡(1305)=(1)(5)−(3)(0)=5−0=5M_{21} = \det \begin{pmatrix} 1 & 3 \\ 0 & 5 \end{pmatrix} = (1)(5) - (3)(0) = 5 - 0 = 5 M22=det⁡(1305)=(1)(5)−(3)(0)=5−0=5M_{22} = \det \begin{pmatrix} 1 & 3 \\ 0 & 5 \end{pmatrix} = (1)(5) - (3)(0) = 5 - 0 = 5 M23=det⁡(1100)=(1)(0)−(1)(0)=0−0=0M_{23} = \det \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} = (1)(0) - (1)(0) = 0 - 0 = 0 M31=det⁡(1334)=(1)(4)−(3)(3)=4−9=−5M_{31} = \det \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix} = (1)(4) - (3)(3) = 4 - 9 = -5 M32=det⁡(1304)=(1)(4)−(3)(0)=4−0=4M_{32} = \det \begin{pmatrix} 1 & 3 \\ 0 & 4 \end{pmatrix} = (1)(4) - (3)(0) = 4 - 0 = 4 M33=det⁡(1103)=(1)(3)−(1)(0)=3−0=3M_{33} = \det \begin{pmatrix} 1 & 1 \\ 0 & 3 \end{pmatrix} = (1)(3) - (1)(0) = 3 - 0 = 3

    The matrix of minors for B is:

    MB=(1500550−543)\boxed{M_B = \begin{pmatrix} 15 & 0 & 0 \\ 5 & 5 & 0 \\ -5 & 4 & 3 \end{pmatrix}}

  2. Calculate the cofactors Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}.

    C11=(−1)1+1(15)=15C_{11} = (-1)^{1+1}(15) = 15 C12=(−1)1+2(0)=0C_{12} = (-1)^{1+2}(0) = 0 C13=(−1)1+3(0)=0C_{13} = (-1)^{1+3}(0) = 0 C21=(−1)2+1(5)=−5C_{21} = (-1)^{2+1}(5) = -5 C22=(−1)2+2(5)=5C_{22} = (-1)^{2+2}(5) = 5 C23=(−1)2+3(0)=0C_{23} = (-1)^{2+3}(0) = 0 C31=(−1)3+1(−5)=−5C_{31} = (-1)^{3+1}(-5) = -5 C32=(−1)3+2(4)=−4C_{32} = (-1)^{3+2}(4) = -4 C33=(−1)3+3(3)=3C_{33} = (-1)^{3+3}(3) = 3

    The matrix of cofactors for B is:

    CB=(1500−550−5−43)\boxed{C_B = \begin{pmatrix} 15 & 0 & 0 \\ -5 & 5 & 0 \\ -5 & -4 & 3 \end{pmatrix}}
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