Let M_ij and C_ij represents the minor and co–factor of an element, where i and j represent the row and column. 

The minor of the matrix can be obtained for a particular element by removing the row and column where the element is present. 

Then finding the absolute value of the matrix newly formed. 

Also, 

C_ij = (–1)^i+j × M_ij

 A = \(\begin{bmatrix} 1&-3 & 2 \\[0.3em] 4 & -1 & 2 \\[0.3em] 3 & 5 & 2 \end{bmatrix}\)

⇒ M₁₁ = \(\begin{bmatrix} -1&2 \\[0.3em] 5 & 2 \\[0.3em] \end{bmatrix}\)

M₁₁ = –1 × 2 – 5 × 2 

M₁₁ = –12

 ⇒ M₂₁ = \(\begin{bmatrix} -3&2 \\[0.3em] 5 & 2 \\[0.3em] \end{bmatrix}\)

M₂₁ = –3 × 2 – 5 × 2 

M₂₁ = –16

⇒ M₃₁ = \(\begin{bmatrix} -3&2 \\[0.3em] -1 & 2 \\[0.3em] \end{bmatrix}\) 

M₃₁ = –3 × 2 – (–1) × 2

M₃₁ = – 4 

C₁₁ = (–1)¹⁺¹ × M₁₁ 

= 1 × –12 

= –12 

C₂₁ = (–1)²⁺¹ × M₂₁ 

= –1 × –16 

= 16 

C₃₁ = (–1)³⁺¹ × M₃₁ 

= 1 × –4 

= –4

Now expanding along the first column we get,

|A| = a₁₁ × C₁₁ + a₂₁× C₂₁+ a₃₁× C₃₁ 

= 1× (–12) + 4 × 16 + 3× (–4) 

= –12 + 64 –12 

= 40