[Baglama] Compute the determinant of the matrix. beginbmatrix 5&1&3 1&-2&2 3&5&4endbmatrix

Expand the determinant along the first row using the cofactor expansion method. This involves multiplying each element in the first row by its corresponding cofactor and summing the results. The formula is:
$$det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$$$det(A)=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$$, where $$a_{ij}$$$$a_{ij}$$ are the elements of the matrix and $$C_{ij}$$$$C_{ij}$$ are their cofactors.

$$det\begin{bmatrix} 5&1&3\\ 1&-2&2\\ 3&5&4\end{bmatrix} = 5((-2)(4) - (2)(5)) - 1((1)(4) - (2)(3)) + 3((1)(5) - (-2)(3))$$$$det\left[\begin{array}{ccc}5& 1& 3\\ 1& -2& 2\\ 3& 5& 4\end{array}\right]=5((-2)(4)-(2)(5))-1((1)(4)-(2)(3))+3((1)(5)-(-2)(3))$$

Simplify the terms within the parentheses. This involves performing the multiplications and subtractions within each cofactor.

$$5(-8 - 10) - 1(4 - 6) + 3(5 + 6) = 5(-18) - 1(-2) + 3(11)$$$$5(-8-10)-1(4-6)+3(5+6)=5(-18)-1(-2)+3(11)$$

Perform the multiplications. This involves multiplying each element in the first row by its simplified cofactor.

$$5(-18) - 1(-2) + 3(11) = -90 + 2 + 33$$$$5(-18)-1(-2)+3(11)=-90+2+33$$

Perform the additions and subtractions. This involves summing the results from Step 3.

$$-90 + 2 + 33 = -55$$$$-90+2+33=-55$$

State the final result. The determinant of the matrix is -55.