Questions: Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.) [1 2 5 9]

Transcript text: Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.) \[ \left[\begin{array}{ll} 1 & 2 \\ 5 & 9 \end{array}\right] \]

Solution

Solution Steps

To find the inverse of a matrix, we need to check if the determinant of the matrix is non-zero. If the determinant is zero, the inverse does not exist (DNE). If the determinant is non-zero, we can use the formula for the inverse of a 2x2 matrix.

Step 1: Define the Matrix

We are given the matrix: \[ \begin{bmatrix} 1 & 2 \\ 5 & 9 \end{bmatrix} \]

Step 2: Calculate the Determinant

The determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is calculated as: \[ \text{det} = ad - bc \] For our matrix: \[ \text{det} = (1 \cdot 9) - (2 \cdot 5) = 9 - 10 = -1 \]

Step 3: Check if the Determinant is Non-Zero

Since \(\text{det} = -1 \neq 0\), the inverse of the matrix exists.

Step 4: Calculate the Inverse

The inverse of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by: \[ \text{inverse} = \frac{1}{\text{det}} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] For our matrix: \[ \text{inverse} = \frac{1}{-1} \begin{bmatrix} 9 & -2 \\ -5 & 1 \end{bmatrix} = \begin{bmatrix} -9 & 2 \\ 5 & -1 \end{bmatrix} \]