The expansion of the determinant along the second row is:

Steps to Solve

1. Identify the Elements of the Second Row

The second row of the matrix is:

$$ \begin{bmatrix} -1 & 7 & 3 \end{bmatrix} $$

2. Write the Expansion Formula

The expansion of the determinant along the second row is given by:

$$ \text{det}(A) = a_{21}C_{21} + a_{22}C_{22} + a_{23}C_{23} $$

where ( a_{ij} ) is the element from the matrix, and ( C_{ij} ) is the corresponding cofactor, defined as:

$$ C_{ij} = (-1)^{i+j} \cdot \text{det}(M_{ij}) $$

3. Calculate the Cofactors

• For ( a_{21} = -1 ):

  • Minor matrix ( M_{21} ):

  $$ \begin{bmatrix} 5 & 6 \ 3 & 8 \end{bmatrix} $$

  • Determinant of ( M_{21} ):

  $$ \text{det}(M_{21}) = (5)(8) - (6)(3) = 40 - 18 = 22 $$

  • Therefore, ( C_{21} = (-1)^{2+1} \cdot 22 = -22 ).

• For ( a_{22} = 7 ):

  • Minor matrix ( M_{22} ):

  $$ \begin{bmatrix} 4 & 6 \ 2 & 8 \end{bmatrix} $$

  • Determinant of ( M_{22} ):

  $$ \text{det}(M_{22}) = (4)(8) - (6)(2) = 32 - 12 = 20 $$

  • Therefore, ( C_{22} = (-1)^{2+2} \cdot 20 = 20 ).

• For ( a_{23} = 3 ):

  • Minor matrix ( M_{23} ):

  $$ \begin{bmatrix} 4 & 5 \ 2 & 3 \end{bmatrix} $$

  • Determinant of ( M_{23} ):

  $$ \text{det}(M_{23}) = (4)(3) - (5)(2) = 12 - 10 = 2 $$

  • Therefore, ( C_{23} = (-1)^{2+3} \cdot 2 = -2 ).

4. Combine the Results

Now substitute the values into the expansion formula:

$$ \text{det}(A) = -1(-22) + 7(20) + 3(-2) $$

Calculating each term gives:

$$ \text{det}(A) = 22 + 140 - 6 $$

5. Final Calculation

Combine the results to find the final determinant value:

$$ \text{det}(A) = 22 + 140 - 6 = 156 $$