Find the Determinant of the Resulting Matrix

Problem

det([[4,−7,4],[0,−3,4],[0,−8,9]]−[[4,0,0],[0,4,0],[0,0,4]])

Solution

1. Perform matrix subtraction by subtracting the corresponding elements of the second matrix from the first matrix.

A=[[4−4,−7−0,4−0],[0−0,−3−4,4−0],[0−0,−8−0,9−4]]

A=[[0,−7,4],[0,−7,4],[0,−8,5]]

2. Identify the property of the resulting matrix. Notice that the first column consists entirely of zeros.

(C_1)=[[0],[0],[0]]

3. Calculate the determinant using the property that if any row or column of a square matrix is all zeros, the determinant of that matrix is zero.

det(A)=0*(−7⋅5−4⋅−8)−0*(−7⋅5−4⋅−8)+0*(−7⋅4−4⋅−7)

det(A)=0

Final Answer

det([[4,−7,4],[0,−3,4],[0,−8,9]]−[[4,0,0],[0,4,0],[0,0,4]])=0

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