Question

Let A be an 6×6 matrix with det(A)=6. Let B be the matrix that can be obtained by applying the following row operations to A :

- R5→R5+0.2R1
- R1↔R5
- R2→R2+0.25R3
- R6→5R6
What is det(B)?

Answer 1

To find the determinant of matrix B, apply the given row operations to matrix A and calculate the determinant of the resulting matrix B. The determinant of matrix B, obtained by applying specified row operations to matrix A with det(A) = 6, is -30.

Step-by-step explanation:

You want to know the determinant of matrix B, which is derived from matrix A through a series of row operations and where det(A) = 6.

Let's consider the effects of the following operations:

• R5→R5+0.2R1 - Adding a multiple of one row to another does not change the determinant.
• R1↔R5 - Swapping two rows multiplies the determinant by -1.
• R2→R2+0.25R3 - Again, adding a multiple of one row to another does not change the determinant.
• R6→ 5R6 - Multiplying a row by a scalar multiplies the determinant by that scalar.

So after applying these operations to A, the determinant of B will be:

det(B) = -1 x 5 x det(A) = -1 x 5 x 6 = -30.

In conclusion, the determinant of matrix B after the given row operations will be -30.