Multiple choice

The inverse of the 2 x 2 matrix $\begin{bmatrix} \ 1 & 2 \ \ 5 & 7 \ \end{bmatrix}$ is

  1. $\frac{1}{3}\begin{bmatrix} \ -7 & 2 \\ \ 5 & -1 \\ \end{bmatrix}$
  2. $\frac{1}{3}\begin{bmatrix} \ 7 & 2 \\ \ 5 & 1 \\ \end{bmatrix}$
  3. $\frac{1}{3}\begin{bmatrix} \ 7 & -2 \\ \ -5 & 1 \\ \end{bmatrix}$
  4. $\frac{1}{3}\begin{bmatrix} \ -7 & -2 \\ \ -5 & -1 \\ \end{bmatrix}$
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Explanation

The inverse of a 2x2 matrix [[a, b], [c, d]] is (1/det) * [[d, -b], [-c, a]]. The determinant is (1*7 - 2*5) = 7 - 10 = -3. The inverse is (1/-3) * [[7, -2], [-5, 1]], which simplifies to (1/3) * [[-7, 2], [5, -1]].

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