Find the Determinant

Problem

det(1)

Solution

1. Partition the matrix into a block triangular form. Notice that the bottom-left 3×2 submatrix consists entirely of zeros. This allows us to calculate the determinant as the product of the determinants of the diagonal blocks.

2. Identify the diagonal blocks. The first block A is the top-left 2×2 matrix, and the second block B is the bottom-right 3×3 matrix.

A=[[1,2],[4,9]]

B=[[1,0,−2],[2,1,3],[0,4,−4]]

3. Calculate the determinant of block A

det(A)=(1)*(9)−(2)*(4)

det(A)=9−8=1

4. Calculate the determinant of block B using cofactor expansion along the first column.

det(B)=1⋅det(1)−2⋅det(0)+0

det(B)=1*((1)*(−4)−(3)*(4))−2*((0)*(−4)−(−2)*(4))

det(B)=1*(−4−12)−2*(0+8)

det(B)=−16−16=−32

5. Multiply the determinants of the blocks to find the total determinant.

det(M)=det(A)⋅det(B)

det(M)=1⋅(−32)=−32

Final Answer

det(1)=−32

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