Question

Let A=[[a,b,c],[d,e,f],[g,h,i]] and B=[[-6d,3e,f-7d],[-6a,3b,c-7a],[-6g,3h,i-7g]] If det (A)=-5, what is det (B)?

Answer 1

To calculate det (B) based on det (A) which is -5, we use the property of determinants where scaling a row or column multiplies the determinant by that scalar. Multiplying the factors from the scaling of matrix A's columns to form B gives us det (B) = -18 * (-5) = 90, assuming the operations on the third column do not make the determinant zero.

Step-by-step explanation:

To find the determinant of matrix B, we can start by looking for any patterns in its formation based on the determinant of matrix A, which is given as -5. Observe that matrix B is obtained by multiplying certain elements of matrix A by specific scalars and adding some scaled elements of matrix A to other elements within the same row.

Specifically, each element in the first column of matrix B is -6 times its corresponding element in the first column of matrix A, each element in the second column of matrix B is 3 times the corresponding element in the second column of matrix A. For the third column of matrix B, it's a mix of an unchanged component from matrix A and a subtraction involving another element of matrix A.

To calculate det (B), remember a property of determinants that states the determinant of a matrix is a linear function with respect to each row or column. So, when a row (or column) of a matrix is multiplied by a scalar, the determinant of the modified matrix is the scalar multiplied by the determinant of the original matrix. Moreover, if a matrix has two proportional rows (or columns), then its determinant is zero. But careful examination is needed, as the elements of the third column in matrix B are not simple scalar multiples of those in matrix A.

Considering the linearity property, for the first two columns where scalar multiplication occurs, we can say:

1. The determinant will be affected by -6 for the first column.
2. The determinant will be affected by 3 for the second column.

As a result, det (B) will be -6 * 3 times det (A), assuming the third column doesn't have any element that would nullify the determinant.

Therefore, det (B) = -18 * (-5) = 90.

Note that this assumes the third column transformations don't introduce dependency between the columns resulting in the determinant being zero.