To find the minors and cofactors of the elements of the given determinant: $$ \begin{vmatrix} 1 & 2 & -3 \\ -2 & 0 & 4 \\ 5 & -1 & 3 \end{vmatrix} $$ Let the matrix be $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$. The minor $M_{ij}$ is the determinant of the submatrix formed by deleting row $i$ and column $j$. The cofactor $C_{ij}$ is given by $C_{ij} = (-1)^{i+j} M_{ij}$. For element $a_{11} = 1$: Step 1: Find the minor $M_{11}$. $$ M_{11} = \begin{vmatrix} 0 & 4 \\ -1 & 3 \end{vmatrix} = (0)(3) - (4)(-1) = 0 - (-4) = 4 $$ Step 2: Find the cofactor $C_{11}$. $$ C_{11} = (-1)^{1+1} M_{11} = (1)(4) = 4 $$ For element $a_{12} = 2$: Step 1: Find the minor $M_{12}$. $$ M_{12} = \begin{vmatrix} -2 & 4 \\ 5 & 3 \end{vmatrix} = (-2)(3) - (4)(5) = -6 - 20 = -26 $$ Step 2: Find the cofactor $C_{12}$. $$ C_{12} = (-1)^{1+2} M_{12} = (-1)(-26) = 26 $$ For element $a_{13} = -3$: Step 1: Find the minor $M_{13}$. $$ M_{13} = \begin{vmatrix} -2 & 0 \\ 5 & -1 \end{vmatrix} = (-2)(-1) - (0)(5) = 2 - 0 = 2 $$ Step 2: Find the cofactor $C_{13}$. $$ C_{13} = (-1)^{1+3} M_{13} = (1)(2) = 2 $$ For element $a_{21} = -2$: Step 1: Find the minor $M_{21}$. $$ M_{21} = \begin{vmatrix} 2 & -3 \\ -1 & 3 \end{vmatrix} = (2)(3) - (-3)(-1) = 6 - 3 = 3 $$ Step 2: Find the cofactor $C_{21}$. $$ C_{21} = (-1)^{2+1} M_{21} = (-1)(3) = -3 $$ For element $a_{22} = 0$: Step 1: Find the minor $M_{22}$. $$ M_{22} = \begin{vmatrix} 1 & -3 \\ 5 & 3 \end{vmatrix} = (1)(3) - (-3)(5) = 3 - (-15) = 18 $$ Step 2: Find the cofactor $C_{22}$. $$ C_{22} = (-1)^{2+2} M_{22} = (1)(18) = 18 $$ For element $a_{23} = 4$: Step 1: Find the minor $M_{23}$. $$ M_{23} = \begin{vmatrix} 1 & 2 \\ 5 & -1 \end{vmatrix} = (1)(-1) - (2)(5) = -1 - 10 = -11 $$ Step 2: Find the cofactor $C_{23}$. $$ C_{23} = (-1)^{2+3} M_{23} = (-1)(-11) = 11 $$ For element $a_{31} = 5$: Step 1: Find the minor $M_{31}$. $$ M_{31} = \begin{vmatrix} 2 & -3 \\ 0 & 4 \end{vmatrix} = (2)(4) - (-3)(0) = 8 - 0 = 8 $$ Step 2: Find the cofactor $C_{31}$. $$ C_{31} = (-1)^{3+1} M_{31} = (1)(8) = 8 $$ For element $a_{32} = -1$: Step 1: Find the minor $M_{32}$. $$ M_{32} = \begin{vmatrix} 1 & -3 \\ -2 & 4 \end{vmatrix} = (1)(4) - (-3)(-2) = 4 - 6 = -2 $$ Step 2: Find the cofactor $C_{32}$. $$ C_{32} = (-1)^{3+2} M_{32} = (-1)(-2) = 2 $$ For element $a_{33} = 3$: Step 1: Find the minor $M_{33}$. $$ M_{33} = \begin{vmatrix} 1 & 2 \\ -2 & 0 \end{vmatrix} = (1)(0) - (2)(-2) = 0 - (-4) = 4 $$ Step 2: Find the cofactor $C_{33}$. $$ C_{33} = (-1)^{3+3} M_{33} = (1)(4) = 4 $$ The minors and cofactors are: $M_{11} = 4$, $C_{11} = 4$ $M_{12} = -26$, $C_{12} = 26$ $M_{13} = 2$, $C_{13} = 2$ $M_{21} = 3$, $C_{21} = -3$ $M_{22} = 18$, $C_{22} = 18$ $M_{23} = -11$, $C_{23} = 11$ $M_{31} = 8$, $C_{31} = 8$ $M_{32} = -2$, $C_{32} = 2$ $M_{33} = 4$, $C_{33} = 4$ The matrix of minors is: $$ \begin{bmatrix} 4 & -26 & 2 \\ 3 & 18 & -11 \\ 8 & -2 & 4 \end{bmatrix} $$ The matrix of cofactors is: $$ \boxed{\begin{bmatrix} 4 & 26 & 2 \\ -3 & 18 & 11 \\ 8 & 2 & 4 \end{bmatrix}} $$