4. What is the determinant of \( A=\left(\begin{array}{ccc}2 & 0 & -2 \\ 1 & 0 & 3 \\ 7 & 0 & 5\end{array}\right) \) ? \( \begin{array}{llll}\text { A. } 20 & \text { B. } 0 & \text { C. } 16 & \text { D. } 24\end{array} \)

To find the determinant of the matrix \( A=\left(\begin{array}{ccc}2 & 0 & -2 \\ 1 & 0 & 3 \\ 7 & 0 & 5\end{array}\right) \), we can use the formula for the determinant of a 3x3 matrix: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Here, \( a=2, b=0, c=-2, d=1, e=0, f=3, g=7, h=0, i=5 \). Plugging in these values, we find: \[ \text{det}(A) = 2(0 \cdot 5 - 3 \cdot 0) - 0(1 \cdot 5 - 3 \cdot 7) + (-2)(1 \cdot 0 - 0 \cdot 7) \] \[ = 2(0) - 0 + (-2)(0) = 0 \] So, the determinant is \( \text{det}(A) = 0 \). The correct answer is \( \text{B. } 0 \). If a determinant is zero, it signifies that the matrix represents a system of linear equations with either infinitely many solutions or no solutions at all—an interesting concept in linear algebra! Additionally, understanding determinants can empower you in various real-world scenarios, such as solving systems of equations in physics and engineering. Determinants can indicate whether a system is stable or unstable, helping in everything from architectural designs to electronic circuits!