如果

$$A = \left [\bestart{matrix}-4 & 0 & -3\-1 & 5 & -1\0 & -3 & 2\end{matrix}\right ]，$$

那么以下哪一个是 \（A^{2}\）？

Question

如果

$$A = \left [\bestart{matrix}-4 & 0 & -3\-1 & 5 & -1\0 & -3 & 2\end{matrix}\right ]，$$

那么以下哪一个是 \（A^{2}\）？

BlackTom AI · Accepted Answer

Compute A^2 by matrix multiplication: A^2 = A · A. Given A = [ [-4, 0, -3]; [-1, 5, -1]; [0, -3, 2] ].

1. Relevant knowledge points and concepts
- Matrix multiplication: (A^2)_{ij} is the dot product of row i of A with column j of A.
- Do not confuse with elementwise squaring; A^2 means A multiplied by itself.

2. Complete step-by-step reasoning and intermediate calculations
Write columns of A for convenience:
col1 = (-4, -1, 0)^T, col2 = (0, 5, -3)^T, col3 = (-3, -1, 2)^T.

Compute each entry (row i · column j):

Row 1 = (-4, 0, -3)
- (1,1): (-4)(-4) + 0(-1) + (-3)(0) = 16 + 0 + 0 = 16
- (1,2): (-4)(0) + 0(5) + (-3)(-3) = 0 + 0 + 9 = 9
- (1,3): (-4)(-3) + 0(-1) + (-3)(2) = 12 + 0 - 6 = 6

Row 2 = (-1, 5, -1)
- (2,1): (-1)(-4) + 5(-1) + (-1)(0) = 4 - 5 + 0 = -1
- (2,2): (-1)(0) + 5(5) + (-1)(-3) = 0 + 25 + 3 = 28
- (2,3): (-1)(-3) + 5(-1) + (-1)(2) = 3 - 5 - 2 = -4

Row 3 = (0, -3, 2)
- (3,1): 0(-4) + (-3)(-1) + 2(0) = 0 + 3 + 0 = 3
- (3,2): 0(0) + (-3)(5) + 2(-3) = 0 -15 -6 = -21
- (3,3): 0(-3) + (-3)(-1) + 2(2) = 0 + 3 + 4 = 7

Thus A^2 = [ [16, 9, 6]; [-1, 28, -4]; [3, -21, 7] ].

3. Why the correct answer is correct
Each entry is obtained by the standard dot-product rule for matrix multiplication; the computed nine values match exactly the resulting product A·A.

4. Why other options would be incorrect
Common mistakes include elementwise squaring (which would give [16,0,9;1,25,1;0,9,4], not valid here) or computing A^T A or A A^T (different results). Any answer not equal to the dot-product results above is incorrect.

Conclusion: The correct A^2 is the matrix [[16, 9, 6]; [-1, 28, -4]; [3, -21, 7]], which matches the provided answer.

如果 \[A = \left [\bestart{matrix}-4 & 0 & -3\\-1 & 5 & -1\\0 & -3 & 2\end{matrix}\right ]，\] 那么以下哪一个是 \（A^{2}\）？单项选择题

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