1. The problem is to understand the matrix $A$ provided: $$A = \begin{bmatrix} 1 & 0 & -2 \\ 4 & 2 & 7 \\ 1 & -5 & 4 \end{bmatrix}$$ 2. Since the task does not specify what operation or analysis to perform on matrix $A$, let's clarify some basic properties. 3. We can interpret $A$ as a $3 \times 3$ matrix, which is square and can be analyzed for determinant, inverse, eigenvalues, etc. 4. Let's compute the determinant of $A$ to determine if $A$ is invertible. 5. Recall the formula for the determinant of a $3 \times 3$ matrix: $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix elements are: $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ 6. From $A$: $$a=1, b=0, c=-2, d=4, e=2, f=7, g=1, h=-5, i=4$$ 7. Substitute: $$\det(A) = 1(2 \times 4 - 7 \times (-5)) - 0(4 \times 4 - 7 \times 1) + (-2)(4 \times (-5) - 2 \times 1)$$ 8. Calculate each term: $$1(8 + 35) - 0(16 - 7) + (-2)(-20 - 2) = 1(43) - 0 + (-2)(-22) = 43 + 44 = 87$$ 9. Since $\det(A) = 87 \neq 0$, matrix $A$ is invertible. Final answer: The determinant of $A$ is $87$, so $A$ is invertible.