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To find the minors and cofactors of the elements of the given determinant:

Mathematics

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To find the minors and cofactors of the elements of the given determinant:

∣12−3−2045−13∣\begin{vmatrix} 1 & 2 & -3 \\ -2 & 0 & 4 \\ 5 & -1 & 3 \end{vmatrix}

Let the matrix be A=[a11a12a13a21a22a23a31a32a33]A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}.
The minor MijM_{ij} is the determinant of the submatrix formed by deleting row ii and column jj.
The cofactor CijC_{ij} is given by Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}.

For element a11=1a_{11} = 1:

  1. Find the minor M11M_{11}.

    M11=∣04−13∣=(0)(3)−(4)(−1)=0−(−4)=4M_{11} = \begin{vmatrix} 0 & 4 \\ -1 & 3 \end{vmatrix} = (0)(3) - (4)(-1) = 0 - (-4) = 4
  2. Find the cofactor C11C_{11}.

    C11=(−1)1+1M11=(1)(4)=4C_{11} = (-1)^{1+1} M_{11} = (1)(4) = 4

    For element a12=2a_{12} = 2:

  3. Find the minor M12M_{12}.

    M12=∣−2453∣=(−2)(3)−(4)(5)=−6−20=−26M_{12} = \begin{vmatrix} -2 & 4 \\ 5 & 3 \end{vmatrix} = (-2)(3) - (4)(5) = -6 - 20 = -26
  4. Find the cofactor C12C_{12}.

    C12=(−1)1+2M12=(−1)(−26)=26C_{12} = (-1)^{1+2} M_{12} = (-1)(-26) = 26

    For element a13=−3a_{13} = -3:

  5. Find the minor M13M_{13}.

    M13=∣−205−1∣=(−2)(−1)−(0)(5)=2−0=2M_{13} = \begin{vmatrix} -2 & 0 \\ 5 & -1 \end{vmatrix} = (-2)(-1) - (0)(5) = 2 - 0 = 2
  6. Find the cofactor C13C_{13}.

    C13=(−1)1+3M13=(1)(2)=2C_{13} = (-1)^{1+3} M_{13} = (1)(2) = 2

    For element a21=−2a_{21} = -2:

  7. Find the minor M21M_{21}.

    M21=∣2−3−13∣=(2)(3)−(−3)(−1)=6−3=3M_{21} = \begin{vmatrix} 2 & -3 \\ -1 & 3 \end{vmatrix} = (2)(3) - (-3)(-1) = 6 - 3 = 3
  8. Find the cofactor C21C_{21}.

    C21=(−1)2+1M21=(−1)(3)=−3C_{21} = (-1)^{2+1} M_{21} = (-1)(3) = -3

    For element a22=0a_{22} = 0:

  9. Find the minor M22M_{22}.

    M22=∣1−353∣=(1)(3)−(−3)(5)=3−(−15)=18M_{22} = \begin{vmatrix} 1 & -3 \\ 5 & 3 \end{vmatrix} = (1)(3) - (-3)(5) = 3 - (-15) = 18
  10. Find the cofactor C22C_{22}.

    C22=(−1)2+2M22=(1)(18)=18C_{22} = (-1)^{2+2} M_{22} = (1)(18) = 18

    For element a23=4a_{23} = 4:

  11. Find the minor M23M_{23}.

    M23=∣125−1∣=(1)(−1)−(2)(5)=−1−10=−11M_{23} = \begin{vmatrix} 1 & 2 \\ 5 & -1 \end{vmatrix} = (1)(-1) - (2)(5) = -1 - 10 = -11
  12. Find the cofactor C23C_{23}.

    C23=(−1)2+3M23=(−1)(−11)=11C_{23} = (-1)^{2+3} M_{23} = (-1)(-11) = 11

    For element a31=5a_{31} = 5:

  13. Find the minor M31M_{31}.

    M31=∣2−304∣=(2)(4)−(−3)(0)=8−0=8M_{31} = \begin{vmatrix} 2 & -3 \\ 0 & 4 \end{vmatrix} = (2)(4) - (-3)(0) = 8 - 0 = 8
  14. Find the cofactor C31C_{31}.

    C31=(−1)3+1M31=(1)(8)=8C_{31} = (-1)^{3+1} M_{31} = (1)(8) = 8

    For element a32=−1a_{32} = -1:

  15. Find the minor M32M_{32}.

    M32=∣1−3−24∣=(1)(4)−(−3)(−2)=4−6=−2M_{32} = \begin{vmatrix} 1 & -3 \\ -2 & 4 \end{vmatrix} = (1)(4) - (-3)(-2) = 4 - 6 = -2
  16. Find the cofactor C32C_{32}.

    C32=(−1)3+2M32=(−1)(−2)=2C_{32} = (-1)^{3+2} M_{32} = (-1)(-2) = 2

    For element a33=3a_{33} = 3:

  17. Find the minor M33M_{33}.

    M33=∣12−20∣=(1)(0)−(2)(−2)=0−(−4)=4M_{33} = \begin{vmatrix} 1 & 2 \\ -2 & 0 \end{vmatrix} = (1)(0) - (2)(-2) = 0 - (-4) = 4
  18. Find the cofactor C33C_{33}.

    C33=(−1)3+3M33=(1)(4)=4C_{33} = (-1)^{3+3} M_{33} = (1)(4) = 4

    The minors and cofactors are:

    • M11=4M_{11} = 4, C11=4C_{11} = 4
    • M12=−26M_{12} = -26, C12=26C_{12} = 26
    • M13=2M_{13} = 2, C13=2C_{13} = 2
    • M21=3M_{21} = 3, C21=−3C_{21} = -3
    • M22=18M_{22} = 18, C22=18C_{22} = 18
    • M23=−11M_{23} = -11, C23=11C_{23} = 11
    • M31=8M_{31} = 8, C31=8C_{31} = 8
    • M32=−2M_{32} = -2, C32=2C_{32} = 2
    • M33=4M_{33} = 4, C33=4C_{33} = 4

    The matrix of minors is:

    [4−262318−118−24]\begin{bmatrix} 4 & -26 & 2 \\ 3 & 18 & -11 \\ 8 & -2 & 4 \end{bmatrix}

    The matrix of cofactors is:

    [4262−31811824]\boxed{\begin{bmatrix} 4 & 26 & 2 \\ -3 & 18 & 11 \\ 8 & 2 & 4 \end{bmatrix}}
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