4. Find the inverse matrix for the given matrix. \[ \begin{array}{l} {\left[\begin{array}{ll} -4 & -5 \\ -1 & -2 \end{array}\right]} \\ A^{-1}=? \end{array} \] $\left[\begin{array}{cc}\frac{-2}{3} & \frac{2}{3} \\ \frac{1}{3} & \frac{-2}{3}\end{array}\right]$ $\left[\begin{array}{cc}\frac{-2}{3} & \frac{5}{3} \\ \frac{1}{3} & \frac{-4}{3}\end{array}\right]$ $\left[\begin{array}{cc}\frac{0}{5} & \frac{-3}{5} \\ \frac{6}{5} & \frac{-4}{5}\end{array}\right]$ $\left[\begin{array}{cc}\frac{-1}{5} & \frac{-3}{5} \\ \frac{7}{5} & \frac{-4}{5}\end{array}\right]$ This matrix has no inverse.

Step 1 ：Given the matrix A = \(\begin{bmatrix} -4 & -5 \\ -1 & -2 \end{bmatrix}\)

Step 2 ：To find the inverse of a matrix, we first need to calculate the determinant of the matrix. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by the formula \(ad - bc\).

Step 3 ：Substituting the values from matrix A into the formula, we get the determinant as -4*(-2) - (-5)*(-1) = -3.

Step 4 ：If the determinant is not zero, then the matrix has an inverse. In this case, the determinant is -3, which is not zero, so the matrix does have an inverse.

Step 5 ：The formula for the inverse of a 2x2 matrix is \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)

Step 6 ：Substituting the values from matrix A and the determinant into the formula, we get the inverse matrix as \(A^{-1} = \frac{1}{-3} \begin{bmatrix} -2 & 5 \\ 1 & -4 \end{bmatrix} = \begin{bmatrix} 0.66666667 & -1.66666667 \\ -0.33333333 & 1.33333333 \end{bmatrix}\)

Step 7 ：Final Answer: The inverse of the given matrix is \(\boxed{\begin{bmatrix} 0.66666667 & -1.66666667 \\ -0.33333333 & 1.33333333 \end{bmatrix}}\)