Questions: Linear Algebra: Matrix trix, A=[ 1 1 0 0 0 ... 0 -1 1 0 0 0 ... 0 -1 0 1 0 0 ... 0 -1 0 0 1 0 ... 0 -1 0 0 0 1 ... 0 -1 0 ... ... ... 0 0 -1 0 ... ... ... 1 0 -1 0 0 0 0 ... 1 ] What is det A ?

Transcript text: Linear Algebra: Matrix \[ \text { trix, } A=\left[\begin{array}{ccccccc} 1 & 1 & 0 & 0 & 0 & \ldots & 0 \\ -1 & 1 & 0 & 0 & 0 & \ldots & 0 \\ -1 & 0 & 1 & 0 & 0 & \ldots & 0 \\ -1 & 0 & 0 & 1 & 0 & \ldots & 0 \\ -1 & 0 & 0 & 0 & 1 & \ldots & 0 \\ -1 & 0 & \ldots & \ldots & \ldots & 0 & 0 \\ -1 & 0 & \ldots & \ldots & \ldots & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 & \ldots & 1 \end{array}\right] \] What is $\operatorname{det} A$ ?

Solution

Solution Steps

To find the determinant of the given matrix $ A $, we can use Python's NumPy library, which provides a convenient function to compute the determinant of a matrix. The matrix $ A $ is a square matrix, so we can directly use the numpy.linalg.det function to find its determinant.

Step 1: Define the Matrix $ A $

The given matrix $ A $ is: \[ A = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]

Step 2: Calculate the Determinant

To find the determinant of matrix $ A $, we use the determinant formula for square matrices. The determinant of matrix $ A $ is calculated to be: \[ \operatorname{det}(A) = 2.0 \]

Final Answer

The determinant of matrix $ A $ is: \[ \boxed{2.0} \]

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