Simplify the Matrix det [[12,13,14],[11,12,13],[9,6,3]]

Problem

det(12)

Solution

1. Apply row operations to simplify the matrix. Subtract the second row from the first row ((R_1)→(R_1)−(R_2) to create smaller numbers.

det(1)

2. Factor out a common factor from the third row. Notice that 9, 6, and 3 are all divisible by 3.

3*det(1)

3. Perform further row operations to create zeros. Subtract 11 times the first row from the second row ((R_2)→(R_2)−11*(R_1) and 3 times the first row from the third row ((R_3)→(R_3)−3*(R_1).

3*det(1)

4. Observe the rows of the resulting matrix. The third row is a multiple of the second row ((R_3)=−1⋅(R_2).

(R_3)+(R_2)=[[0,0,0]]

5. Conclude the determinant value. Since the matrix has a row of zeros after row operations (or because two rows are linearly dependent), the determinant is 0.

3⋅0=0

Final Answer

det(12)=0

Want more problems? Check here!