Find the Determinant

Problem

det(a)

Solution

1. Identify the structure of the matrix as a block diagonal matrix.

2. Recognize that the determinant of a block diagonal matrix is the product of the determinants of its diagonal blocks.

3. Define the two 2×2 diagonal blocks as (M_1) and (M_2)

(M_1)=[[a,b],[c,d]]

(M_2)=[[a,−b],[c,d]]

4. Calculate the determinant of the first block (M_1)

det((M_1))=a*d−b*c

5. Calculate the determinant of the second block (M_2)

det((M_2))=a(d)−(−b)*(c)=a*d+b*c

6. Multiply the two determinants together to find the determinant of the original 4×4 matrix.

det(M)=(a*d−b*c)*(a*d+b*c)

7. Simplify the product using the difference of squares formula.

(a*d−b*c)*(a*d+b*c)=(a*d)2−(b*c)2

(a*d)2−(b*c)2=a2*d2−b2*c2

Final Answer

det(a)=a2*d2−b2*c2

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