Let $$ A $$ and $$ B $$ be $$ 3 imes 3 $$ matrices, with $$ \operatorname{det} A=5 $$ and det $$ B=4 $$. Use properties of determinants to complete parts (a) through (e) below. a. Compute det $$ A B $$. det $$ A B=20 $$ (Type an integer or a fraction.) b. Compute det $$ 5 A $$. det $$ 5 A=625 $$ (Type an integer or a fraction.) c. Compute det $$ B^{ op} $$. det $$ B^{ op}=4 $$ (Type an integer or a fraction.) d. Compute det $$ A^{-1} $$. det $$ A^{-1}=\frac{1}{5} $$ (Type an integer or a simplified fraction.) e. Compute det $$ A^{3} $$, det $$ A^{3}=\square $$ (Type an integer or a fraction.)

Question

Let $$ A $$ and $$ B $$ be $$ 3 	imes 3 $$ matrices, with $$ \operatorname{det} A=5 $$ and det $$ B=4 $$. Use properties of determinants to complete parts (a) through (e) below. a. Compute det $$ A B $$. det $$ A B=20 $$ (Type an integer or a fraction.) b. Compute det $$ 5 A $$. det $$ 5 A=625 $$ (Type an integer or a fraction.) c. Compute det $$ B^{	op} $$. det $$ B^{	op}=4 $$ (Type an integer or a fraction.) d. Compute det $$ A^{-1} $$. det $$ A^{-1}=\frac{1}{5} $$ (Type an integer or a simplified fraction.) e. Compute det $$ A^{3} $$, det $$ A^{3}=\square $$ (Type an integer or a fraction.)

UpStudy AI · Accepted Answer

Let $$ A $$ and $$ B $$ be $$ 3 \times 3 $$ matrices with $$\det A = 5$$ and $$\det B = 4$$.

1. For two $$ n \times n $$ matrices, the determinant of their product is:
   $$
   \det(AB) = \det A \cdot \det B.
   $$
   Thus,
   $$
   \det(AB) = 5 \times 4 = 20.
   $$

2. When a matrix is multiplied by a scalar, each row (or column) of the matrix is multiplied by that scalar. Specifically, for a $$ 3 \times 3 $$ matrix, 
   $$
   \det(5A) = 5^3 \cdot \det A.
   $$
   Therefore,
   $$
   \det(5A) = 125 \times 5 = 625.
   $$

3. The determinant of a matrix is equal to the determinant of its transpose:
   $$
   \det(B^\top) = \det B = 4.
   $$

4. The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix:
   $$
   \det(A^{-1}) = \frac{1}{\det A} = \frac{1}{5}.
   $$

5. For the power of a matrix, the determinant is raised to that power:
   $$
   \det(A^3) = (\det A)^3.
   $$
   Since $$\det A = 5$$,
   $$
   \det(A^3) = 5^3 = 125.
   $$

Thus, the answers are:
- (a) $$ \det AB = 20 $$
- (b) $$ \det(5A)=625 $$
- (c) $$ \det(B^\top)=4 $$
- (d) $$ \det(A^{-1})=\frac{1}{5} $$
- (e) $$ \det(A^3)=125 $$
det $$ A B = 20 $$ det $$ 5A = 625 $$ det $$ B^{\top} = 4 $$ det $$ A^{-1} = \frac{1}{5} $$ det $$ A^{3} = 125 $$

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