If

𝐴
=
[
−
4

−
3

−
2

5

2

−
5

−
1

−
2

−
2
]
,

then which of the following is
𝐴
2
?

Question

If

𝐴
=
[
−
4
	
−
3
	
−
2

5
	
2
	
−
5

−
1
	
−
2
	
−
2
]
,

then which of the following is 
𝐴
2
?

BlackTom AI · Accepted Answer

Matrix multiplication uses the row-by-column dot product, so to compute A^2 we multiply A by A and compute each (i,j) entry as the dot product of row i of A with column j of A.

Let A = 
[-4  -3  -2
  5   2  -5
 -1  -2  -2]

Columns of A: c1 = [-4, 5, -1], c2 = [-3, 2, -2], c3 = [-2, -5, -2].

Compute entries of A^2 = A·A:

Row 1 · c1: (1,1) = (-4)(-4) + (-3)(5) + (-2)(-1) = 16 - 15 + 2 = 3.
Row 1 · c2: (1,2) = (-4)(-3) + (-3)(2) + (-2)(-2) = 12 - 6 + 4 = 10.
Row 1 · c3: (1,3) = (-4)(-2) + (-3)(-5) + (-2)(-2) = 8 + 15 + 4 = 27.

Row 2 · c1: (2,1) = 5(-4) + 2(5) + (-5)(-1) = -20 + 10 + 5 = -5.
Row 2 · c2: (2,2) = 5(-3) + 2(2) + (-5)(-2) = -15 + 4 + 10 = -1.
Row 2 · c3: (2,3) = 5(-2) + 2(-5) + (-5)(-2) = -10 - 10 + 10 = -10.

Row 3 · c1: (3,1) = (-1)(-4) + (-2)(5) + (-2)(-1) = 4 - 10 + 2 = -4.
Row 3 · c2: (3,2) = (-1)(-3) + (-2)(2) + (-2)(-2) = 3 - 4 + 4 = 3.
Row 3 · c3: (3,3) = (-1)(-2) + (-2)(-5) + (-2)(-2) = 2 + 10 + 4 = 16.

Assembled, A^2 = 
[  3   10   27
  -5   -1  -10
  -4    3   16 ]

Why this is correct: every entry was computed by the standard matrix product rule and all arithmetic was shown; the resulting matrix matches the given correct answer.

Why other options would be wrong: common mistakes include element-wise squaring (squaring each entry individually), using row-by-row instead of row-by-column, or sign/arithmetic errors in computing dot products. Any answer differing from the matrix above is therefore incorrect.

Final confirmation: The computed A^2 equals [3  10  27; -5  -1  -10; -4  3  16], matching the provided correct answer.

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

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