Questions: Find the inverse of the given matrix, if it exists. M=[ 4 4 1 3 ] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. M^(-1)=[ ] (Simplify your answer. Type an integer or a fraction.) B. The inverse matrix does not exist.

$Find the inverse of the given matrix, if it exists. M=[ 4 4 1 3 ] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. M^(-1)=[ ] (Simplify your answer. Type an integer or a fraction.) B. The inverse matrix does not exist.$

Transcript text: Find the inverse of the given matrix, if it exists. \[ M=\left[\begin{array}{ll} 4 & 4 \\ 1 & 3 \end{array}\right] \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. $\square$ $\square$ $\mathrm{M}^{-1}=\left[\begin{array}{c}\square \\ \square \\ \square\end{array}\right]$ (Simplify your answer. Type an integer or a fraction.) $\square$ $\square$ B. The inverse matrix does not exist.

Solution

Solution Steps

To find the inverse of a given 2x2 matrix, we can use the formula for the inverse of a 2x2 matrix. The matrix $ M $ is given by: \[ M=\left[\begin{array}{ll} 4 & 4 \\ 1 & 3 \end{array}\right] \] The inverse of a 2x2 matrix $ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right] $ is given by: \[ M^{-1} = \frac{1}{ad - bc} \left[\begin{array}{ll} d & -b \\ -c & a \end{array}\right] \] First, we need to calculate the determinant $ ad - bc $. If the determinant is non-zero, we can then use the formula to find the inverse.

Solution Approach

Calculate the determinant of the matrix $ M $.
If the determinant is non-zero, use the formula to find the inverse matrix.
If the determinant is zero, the inverse does not exist.

Step 1: Calculate the Determinant

To find the inverse of the matrix $ M = \left[\begin{array}{ll} 4 & 4 \\ 1 & 3 \end{array}\right] $, we first calculate the determinant using the formula: \[ \text{det}(M) = ad - bc = (4)(3) - (4)(1) = 12 - 4 = 8 \] Thus, the determinant is $ \text{det}(M) = 8 $.

Step 2: Check the Determinant

Since the determinant $ \text{det}(M) $ is non-zero ($ 8 \neq 0 $), the inverse of the matrix exists.

Step 3: Calculate the Inverse

Using the formula for the inverse of a 2x2 matrix: \[ M^{-1} = \frac{1}{\text{det}(M)} \left[\begin{array}{ll} d & -b \\ -c & a \end{array}\right] \] we substitute $ a = 4 $, $ b = 4 $, $ c = 1 $, and $ d = 3 $: \[ M^{-1} = \frac{1}{8} \left[\begin{array}{ll} 3 & -4 \\ -1 & 4 \end{array}\right] = \left[\begin{array}{ll} \frac{3}{8} & -\frac{1}{2} \\ -\frac{1}{8} & \frac{1}{2} \end{array}\right] \]