The multiplicative inkerse of the matrix \( \left[\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right] \) is

To find the multiplicative inverse of the matrix \( \left[\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right] \), we can use the formula for the inverse of a 2x2 matrix, which is given by: \[ A^{-1} = \frac{1}{ad - bc} \left[\begin{array}{ll} d & -b \\ -c & a\end{array}\right] \] where \( A = \left[\begin{array}{ll} a & b \\ c & d\end{array}\right] \). In this case: - \( a = 3 \) - \( b = 5 \) - \( c = 1 \) - \( d = 2 \) First, we calculate the determinant: \[ ad - bc = (3)(2) - (5)(1) = 6 - 5 = 1 \] Now, we can plug these values into the inverse formula: \[ A^{-1} = \frac{1}{1} \left[\begin{array}{ll} 2 & -5 \\ -1 & 3\end{array}\right] = \left[\begin{array}{ll} 2 & -5 \\ -1 & 3\end{array}\right] \] So, the multiplicative inverse of the matrix \( \left[\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right] \) is \( \left[\begin{array}{ll} 2 & -5 \\ -1 & 3\end{array}\right] \). Isn't it fascinating how two simple numbers can unlock so many possibilities in mathematics? Just think, with that inverse matrix in hand, you can solve simultaneous equations or transform geometric shapes—like magic! Next time you're faced with matrices, remember: practice makes perfect! Always double-check your determinant—it's the key that determines whether or not your matrix has an inverse. A slip in those calculations can lead you down the wrong path, but with a little patience, you'll master this skill!