Find the Kernel

Problem

det(d+j*t)=(1−t)*det(d)

Solution

1. Identify the task as finding the determinant of the 3×3 matrix on the left side and comparing it to the right side to verify the identity or solve for the expression.

2. Apply row operations to simplify the determinant on the left. Subtract the second row from the first row: (R_1)→(R_1)−(R_2)

det(d+j*t−(d+j))

3. Simplify the entries in the first row.

det(j*(t−1))

4. Factor out the common term (t−1) from the first row using the linearity property of determinants.

(t−1)*det(j)

5. Apply row operations again to the second row: (R_2)→(R_2)−(R_1)

(t−1)*det(j)

6. Swap rows (R_1) and (R_2) to match the target matrix. Swapping two rows multiplies the determinant by −1

−(t−1)*det(d)

7. Distribute the negative sign into the binomial (t−1)

(1−t)*det(d)

Final Answer

det(d+j*t)=(1−t)*det(d)

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