Determine if \( \left[\begin{array}{rr}-5 & -4 \\ 0 & 1\end{array}\right] \) and \( \left[\begin{array}{cc}4 / 19 & 3 / 19 \\ -5 / 19 & 1 / 19\end{array}\right] \) are inverses of each other.

To determine if the two matrices \[ A = \left[\begin{array}{rr}-5 & -4 \\ 0 & 1\end{array}\right] \] and \[ B = \left[\begin{array}{cc} \frac{4}{19} & \frac{3}{19} \\ -\frac{5}{19} & \frac{1}{19} \end{array}\right] \] are inverses of each other, we need to check if the product \( AB \) equals the identity matrix \( I \). The identity matrix \( I \) for a \( 2 \times 2 \) matrix is \[ I = \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]. \] Now, we will compute the product \( AB \): \[ AB = \left[\begin{array}{rr}-5 & -4 \\ 0 & 1\end{array}\right] \left[\begin{array}{cc} \frac{4}{19} & \frac{3}{19} \\ -\frac{5}{19} & \frac{1}{19} \end{array}\right]. \] To find the elements of the resulting matrix, we will use the formula for matrix multiplication. The element in the first row and first column of \( AB \) is calculated as follows: \[ (-5) \cdot \frac{4}{19} + (-4) \cdot \left(-\frac{5}{19}\right) = -\frac{20}{19} + \frac{20}{19} = 0. \] The element in the first row and second column is: \[ (-5) \cdot \frac{3}{19} + (-4) \cdot \frac{1}{19} = -\frac{15}{19} - \frac{4}{19} = -\frac{19}{19} = -1. \] The element in the second row and first column is: \[ 0 \cdot \frac{4}{19} + 1 \cdot \left(-\frac{5}{19}\right) = 0 - \frac{5}{19} = -\frac{5}{19}. \] The element in the second row and second column is: \[ 0 \cdot \frac{3}{19} + 1 \cdot \frac{1}{19} = 0 + \frac{1}{19} = \frac{1}{19}. \] Putting all these results together, we have: \[ AB = \left[\begin{array}{cc} 0 & -1 \\ -\frac{5}{19} & \frac{1}{19} \end{array}\right]. \] Since \( AB \) does not equal the identity matrix \( I \), we conclude that the matrices \( A \) and \( B \) are not inverses of each other. Thus, the final answer is: \[ \text{No, they are not inverses of each other.} \]