If A is an n times n matrix and det A d what is det kA Plea

If A is an n times n matrix, and det A = d, what is det (kA)? Please justify your answer.

Solution

Let A be a n×n invertible matrix, det(kA)=k^n detA

hence d k^n

When we multiply kA, where A is an n×n matrix, and k a scalar, then every entry aij of matrix A is multiplied by k: i.e. aijkaijkaij for each aij. That means for each row ii 0in, we can factor out k.

Now. the elementary row operations , and how each one of them affects the determinant of the matrix on which it is operating. Specifically, when any one row is multiplied by the scalar k, the determinant of A becomes kdetA. So given that scalar multiplication of a square n×n matrix is equivalent to \"row operating\" on n rows (by multiplying each row by the scalar kk), we can conclude that

det(kA)=kkkntimesdetA=k^n detA