Let $ A = \begin{bmatrix} 2 & -1 \ -3 & -4 \end{bmatrix}$.Which of the following statements is correct?

Question

BlackTom AI · Accepted Answer

We start by restating the problem: given A = [ [2, -1], [-3, -4] ], determine which statement about the inverse is correct. The provided data shows that the claimed inverse is c. A^{-1} = [ [4/11, -1/11], [-3/11, -2/11] ].

First, compute the determinant of A to know if A is invertible. For a 2x2 matrix, det(A) = ad - bc where A = [ [a, b], [c, d] ]. Here a = 2, b = -1, c = -3, d = -4. So det(A) = (2)(-4) - (-1)(-3) = -8 - 3 = -11. Since det(A) ≠ 0, A is invertible, and the inverse exists.

Next, apply the standard formula for the inverse of a 2x2 matrix: A^{-1} = (1/det(A)) * [ [d, -b], [-c, a] ]. Substituting the values gives A^{-1} = (1/(-11)) * [ [-4, 1], [3, 2] ].

Carrying out the scalar multiplication yields A^{-1} = [ [(-4)/(-11), 1/(-11)], [3/(-11), 2/(-11)] ] = [ [4/11, -1/11], [-3/11, -2/11] ].

Thus, the reported inverse in option c matches the correct computation, confirming its accuracy.

Now, regarding the other options: note that the prompt provides only a single option (the one labeled c) and does not supply any alternative choices to evaluate. In a typical multiple-choice setting, one would compare each listed candidate against the actual inverse. Since no other options are given here, there is nothing to refute within this question as presented. The essential check—whether the matrix shown is indeed the inverse—has been performed and confirms correctness.

In summary, the determinant check confirms invertibility, the 2x2 inverse formula yields the matrix [ [4/11, -1/11], [-3/11, -2/11] ], and this matches the stated option c.

Let \( A = \begin{bmatrix} 2 & -1 \\ -3 & -4 \end{bmatrix}\).Which of the following statements is correct?单项选择题

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