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Let A be a 6 -by -6 matrix whose determinant is -1. Use properties of determinants to evaluate the following:

det(2A)= ____

Sagot :

jacob193

Answer:

[tex]\det (2\, A) = (-32)[/tex].

Step-by-step explanation:

If all items in one particular row of an [tex]n \times n[/tex] ([tex]n[/tex] rows) square matrix [tex]A[/tex] are multiplied with a scalar [tex]k[/tex], the determinant of the resultant matrix would be [tex]k[/tex] times the determinant of the original matrix [tex]A[/tex].

To find the determinant when elements in all rows (not just one row) are multiplied with [tex]k[/tex], apply this property iteratively one row at a time. Start by multiplying all elements in the first row [tex]k\, \det(A)[/tex], the next row [tex]k^{2}\, \det(A)[/tex], until reaching the [tex]n[/tex]th row, [tex]k^{n}\, \det(A)[/tex].

Hence, if [tex]A[/tex] is an [tex]n \times n[/tex] matrix, and elements in all rows in [tex]A[/tex] (i.e., all the elements) are multiplied with a scalar [tex]k[/tex], the determinant of the resultant matrix would be [tex]k^{n}[/tex] times the determinant of [tex]A[/tex]:

[tex]\det (k\, A) = k^{n}\, \det(A)[/tex].

In this question:

  • [tex]k = 2[/tex].
  • [tex]n = 6[/tex].
  • [tex]\det (A) = (-1)[/tex].

Therefore:

[tex]\det(2\, A) = 2^{6}\, \det(A) = 32\times (-1) = (-32)[/tex].

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