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

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

UpStudy AI · Accepted Answer

1. We use the property that for an $$ n \times n $$ matrix and a scalar $$ c $$,  
   $$
   \operatorname{det}(cA) = c^n \operatorname{det}(A)
   $$
   Since $$ A $$ is a $$ 3 \times 3 $$ matrix, $$ n=3 $$.

2. Given $$ \operatorname{det} A = 5 $$ and $$ c = 5 $$, we have  
   $$
   \operatorname{det}(5A) = 5^3 \operatorname{det}(A) = 125 \times 5 = 625.
   $$

The final answer is  
$$
\operatorname{det}(5A) = 625.
$$
$$ \operatorname{det}(5A) = 625 $$

Let and be matrices, with and . Use properties of determinants to complete parts (a) through (e) below.
a. Compute det .
(Type an integer or a fraction.)
b. Compute .
(Type an integer or a fraction.)

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Let and be matrices, with and . Use properties of determinants to complete parts (a) through (e) below. a. Compute det . (Type an integer or a fraction.) b. Compute . (Type an integer or a fraction.)