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It looks like the image contains a question asking whether the inverse of a matrix \( A^{-1} \) exists for a given matrix \( A \). The matrix in question is:

\[
A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}
\]

To determine if the inverse of matrix \( A \) exists, we need to check if \( A \) is invertible, which depends on whether its determinant is non-zero.

Let me calculate the determinant of matrix \( A \).

Test whether A^{-1} exists or not for A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}

Check whether the inverse of matrix A exists by calculating the determinant of the matrix A = [1 0 1; 0 1 1; 0 0 1]. This solution provides step-by-step details of how to verify if a matrix is invertible using the determinant rule.

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