Use properties of determinants to evaluate the given...

Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.
$$\left|\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{array}\right|$$

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Key Concepts

Determinant Properties

The determinant is a scalar value that provides important information about a square matrix, such as whether it is invertible and the scaling factor for volume when the matrix is viewed as a transformation. Key properties include linearity in rows or columns and the invariance of the determinant under certain elementary operations, which can be used to simplify calculations.

Row Swapping

Swapping two rows (or columns) of a matrix results in a change of sign of the determinant. This property is crucial for evaluating determinants by inspection, as it allows one to relate the determinant of a modified matrix back to the determinant of a simple or known matrix, often the identity matrix.

Permutation Matrices

Permutation matrices are formed by permuting the rows or columns of an identity matrix. The determinant of a permutation matrix is either +1 or -1, corresponding to the parity of the permutation (even or odd number of swaps). Recognizing a matrix as a permutation of the identity simplifies determining its determinant by understanding its underlying permutation's sign.

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