Let A ∈ℂ^3 ×3 be given by A=[ 1 0 i 0 1 0 -i 0 -1 ] (a) Ca

step 1 Section 3 .2, problem number 30, they give us a matrix A, and they tell us that the determinant

Let A ∈ℂ^3 ×3 be given by A=[ 1 0 i 0 1...

Let $A \in \mathbb{C}^{3 \times 3}$ be given by
$$A=\left[\begin{array}{ccc}1 & 0 & i \\0 & 1 & 0 \\-i & 0 & -1
\end{array}\right]$$
(a) Calculate $\operatorname{det}(A)$.
(b) Find $\operatorname{det}\left(A^{4}\right)$.

Key Concepts

Properties of Determinants with Respect to Matrix Powers

A fundamental property of determinants is that for any square matrix A and any positive integer n, the determinant of the nth power of A is equal to the determinant of A raised to the nth power; mathematically, this is expressed as det(A?) = (det(A))?. This property allows one to efficiently compute determinants of powers of matrices by first computing the determinant of the base matrix.

Computation of 3x3 Determinants

For 3x3 matrices, the determinant is typically calculated using a specific formula that involves summing the products of the elements of the matrix along the main diagonals while subtracting the products along the secondary diagonals. This computation can also be performed by the method of cofactor expansion, and it remains one of the standard techniques taught in linear algebra courses regardless of whether the matrix entries are real or complex.

Determinant of a Matrix

The determinant is a scalar attribute associated with a square matrix that encapsulates key properties of the matrix, such as whether it is invertible, and it relates to the volume scaling factor of the linear transformation defined by the matrix. In the context of complex matrices, the determinant still carries these geometric and algebraic meanings, and its computation follows the same principles as for real matrices.

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