Let A be a 4-by-4 matrix whose determinant is -4. Use properties of determinants to evaluate the following:

Ask by Wade Wood. in the United States

Mar 23,2025

Question

Let A be a 4-by-4 matrix whose determinant is -4. Use properties of determinants to evaluate the following:

Ask by Wade Wood. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

1. We know that for an $$ n \times n $$ matrix and a scalar $$ c $$, 
   $$
   \det(cA) = c^n \det(A).
   $$
   Since $$ A $$ is a $$ 4 \times 4 $$ matrix, we have:
   $$
   \det(2A) = 2^4 \det(A) = 16(-4) = -64.
   $$

2. For the square of a matrix, the property is:
   $$
   \det(A^2) = \left(\det(A)\right)^2.
   $$
   Thus:
   $$
   \det(A^2) = (-4)^2 = 16.
   $$

3. For the inverse of a matrix, the determinant satisfies:
   $$
   \det(A^{-1}) = \frac{1}{\det(A)}.
   $$
   Therefore:
   $$
   \det(A^{-1}) = \frac{1}{-4} = -\frac{1}{4}.
   $$
$$ \det(2A) = -64, \quad \det(A^2) = 16, \quad \det(A^{-1}) = -\frac{1}{4} $$

Let A be a 4-by-4 matrix whose determinant is -4. Use properties of determinants to evaluate the following:

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