Find $| A B |$, if
$$A = \left[ \begin{array} { r r r }
- 3 & 4 & 5 \
- 3 & - 5 & 1 \
5 & - 5 & 1
\end{array} \right] , B = \left[ \begin{array} { r r r }
- 1 & - 2 & 5 \
4 & 1 & - 3 \
- 1 & - 1 & 4
\end{array} \right]$$
A) 2320
B) 2736
C) 202
D) 0
E) $- 2644$

Question

Find $| A B |$, if
$$A = \left[ \begin{array} { r r r } 
- 3 & 4 & 5 \
- 3 & - 5 & 1 \
5 & - 5 & 1
\end{array} \right] , B = \left[ \begin{array} { r r r } 
- 1 & - 2 & 5 \
4 & 1 & - 3 \
- 1 & - 1 & 4
\end{array} \right]$$
A) 2320
B) 2736
C) 202
D) 0
E) $- 2644$

Quizplus · Accepted Answer

Explanation:Given matrices are$$A = \left[ \begin{array} { r r r } - 3 & 4 & 5 \- 3 & - 5 & 1 \5 & - 5 & 1\end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & - 2 & 5 \4 & 1 & - 3 \- 1 & - 1 & 4\end{array} \right]$$To find the determinant of AB, we first multiply the matrices:$$AB = \left[ \begin{array} { r r r } - 3 & 4 & 5 \- 3 & - 5 & 1 \5 & - 5 & 1\end{array} \right] \left[ \begin{array} { r r r } - 1 & - 2 & 5 \4 & 1 & - 3 \- 1 & - 1 & 4\end{array} \right]$$$$= \left[ \begin{array} { r r r } - 3(-1) + 4(4) + 5(-1) & - 3(-2) + 4(1) + 5(-1) & - 3(5) + 4(-3) + 5(4) \- 3(-1) - 5(4) + 1(-1) & - 3(-2) - 5(1) + 1(-1) & - 3(5) - 5(-3) + 1(4) \5(-1) - 5(4) + 1(-1) & 5(-2) - 5(1) + 1(-1) & 5(5) - 5(-3) + 1(4)\end{array} \right]$$$$= \left[ \begin{array} { r r r } 1 & 2 & 1 \- 19 & 11 & 22 \- 21 & - 16 & 32\end{array} \right]$$Now, we find the determinant of AB:$$\left| AB \right| = \left| \begin{array} { r r r } 1 & 2 & 1 \- 19 & 11 & 22 \- 21 & - 16 & 32\end{array} \right|$$$$= 1(11 \times 32 - 22 \times (-16)) - 2(-19 \times 32 - 22 \times (-21)) + 1(-19 \times (-16) - 11 \times (-21))$$$$= 1(352 + 352) - 2(-608 + 462) + 1(304 + 231)$$$$= 1(704) - 2(-146) + 1(535)$$$$= 704 + 292 + 535$$$$= 1531$$Therefore, $| AB | = 1531$.

Find ∣AB∣| A B |∣AB∣ , If A) 2320 B) 2736 C) 202 D) 0 E)

Find $| A B |$ , If A) 2320
B) 2736
C) 202
D) 0
E)