Let

𝐴
=
[
3

5

3

1

−
4

0

−
2

3

1
]
,
and
𝐵
=
[
−
3

4

2

0

1

1

0

−
1

3
]
.

Which of the following options is
−
3
𝐴
𝐵
𝑇
?

Question

Let

𝐴
=
[
3
	
5
	
3

1
	
−
4
	
0

−
2
	
3
	
1
]
,
 and 
𝐵
=
[
−
3
	
4
	
2

0
	
1
	
1

0
	
−
1
	
3
]
.

Which of the following options is 
−
3
𝐴
𝐵
𝑇
?

BlackTom AI · Accepted Answer

Transpose B to obtain B^T = [[-3, 0, 0], [4, 1, -1], [2, 1, 3]]. Key concepts: matrix transpose (swap rows and columns), matrix multiplication (row·column dot products), and scalar multiplication of a matrix.

Step 1 — compute B^T:
B = [[-3, 4, 2],
     [0,  1, 1],
     [0, -1, 3]]
so B^T = [[-3, 0, 0],
          [ 4, 1,-1],
          [ 2, 1, 3]].

Step 2 — compute A B^T by taking dot products of rows of A with columns of B^T. A = [[3,5,3],[1,-4,0],[-2,3,1]].

Calculate entries of C = A B^T:

Row 1 of A = [3, 5, 3]:
- c11 = [3,5,3]·[-3,4,2] = 3(-3)+5(4)+3(2) = -9+20+6 = 17
- c12 = [3,5,3]·[0,1,1]     = 3(0)+5(1)+3(1)  = 0+5+3 = 8
- c13 = [3,5,3]·[0,-1,3]    = 3(0)+5(-1)+3(3) = 0-5+9 = 4
Row 1 of C = [17, 8, 4]

Row 2 of A = [1, -4, 0]:
- c21 = 1(-3)+(-4)(4)+0(2) = -3 -16 + 0 = -19
- c22 = 1(0)+(-4)(1)+0(1)  = 0 -4 + 0 = -4
- c23 = 1(0)+(-4)(-1)+0(3) = 0 +4 + 0 = 4
Row 2 of C = [-19, -4, 4]

Row 3 of A = [-2, 3, 1]:
- c31 = (-2)(-3)+3(4)+1(2) = 6+12+2 = 20
- c32 = (-2)(0)+3(1)+1(1)   = 0+3+1 = 4
- c33 = (-2)(0)+3(-1)+1(3)  = 0-3+3 = 0
Row 3 of C = [20, 4, 0]

Thus AB^T = [[17,8,4],[-19,-4,4],[20,4,0]].

Step 3 — multiply by the scalar -3: -3 · AB^T =
[[ -51, -24, -12],
 [  57,  12, -12],
 [ -60, -12,   0]].

Why this is correct: each entry of the final matrix equals -3 times the corresponding dot product computed for AB^T. Any alternative answer would have to have identical entries; deviations indicate arithmetic or transpose mistakes (common errors include transposing incorrectly or using wrong columns for dot products).

No other answer options were listed; the computed matrix matches the provided correct answer exactly. Confirmed final answer: [[-51, -24, -12], [57, 12, -12], [-60, -12, 0]].

Let 𝐴 = [ 3 5 3 1 − 4 0 − 2 3 1 ] , and 𝐵 = [ − 3 4 2 0 1 1 0 − 1 3 ] . Which of the following options is − 3 𝐴 𝐵 𝑇 ?单项选择题

类似问题

Let \[A = \left [\begin{matrix}-1 & 1 & 3\\-2 & 5 & -4\\-3 & -2 & 3\end{matrix}\right ], \text { and } B = \left [\begin{matrix}0 & -5 & -1\\4 & 1 & 0\\1 & 4 & 2\end{matrix}\right ].\] Which of the following options is \(- 2 A B^{T}\)?

设 \[A = \左 [\begin{matrix}-2 & -2\\-3 & -2\\4 & 4\end{matrix}\right ]， \text { 和} B = \左 [\begin{matrix}-5 & 2 & 1\\2 & -3 & 0\\2 & 4 & 4\4\end{matrix}\right ].\] 以下哪种选项是\（- 3 A B^{T}\）？

If 𝐴 = [ 5 − 5 5 1 − 2 − 1 1 − 4 − 4 ] , then which of the following is 𝐴 2 ?

If 𝐴 = [ − 4 − 3 − 2 5 2 − 5 − 1 − 2 − 2 ] , then which of the following is 𝐴 2 ?

设 A = \左 [\begin{matrix}-2 & -2\\-3 & -2\\4 & 4\end{matrix}\right ]， \text { 和} B = \左 [\begin{matrix}-5 & 2 & 1\\2 & -3 & 0\\2 & 4 & 4\4\end{matrix}\right ]。以下哪种选项是\（- 3 A B^{T}\）？

设 \[A = \左 [\begin{matrix}-3 & 0 & 2\\-4 & 1 & -4\\1 & 2 & -4\end{matrix}\right ]， \text { 和 } B = \left [\begin{matrix}-5 & 1 & -4\\4 & -4\\2 & 2 & 2\end{matrix}\right ].\] 以下哪种选项是\（- 3 A B^{T}\）？

Let 𝐴 = [ − 3 0 2 − 4 1 − 4 1 2 − 4 ] , and 𝐵 = [ − 5 1 − 4 4 4 − 4 2 2 2 ] . Which of the following options is − 3 𝐴 𝐵 𝑇 ?

If 𝐴 = [ − 4 0 − 3 − 1 5 − 1 0 − 3 2 ] , then which of the following is 𝐴 2 ?

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