In each part, find the determinant given that A is a 4 ×4 matrix for which det(A)=-2 (a) det(-A)

step 1 Hi, so let's consider a matrix a square matrix 4 by 4 that has the terminate equals to minus 2.

In each part, find the determinant given that A is a 4 ×4...

In each part, find the determinant given that $A$ is a $4 \times 4$ matrix for which $\operatorname{det}(A)=-2$
(a) $\operatorname{det}(-A)$
(b) $\operatorname{det}\left(A^{-1}\right)$
(c) $\operatorname{det}\left(2 A^{T}\right)$
(d) $\operatorname{det}\left(A^{3}\right)$

Key Concepts

Matrix Determinant Properties

The determinant of a square matrix provides a scalar summary of many useful properties including invertibility and volume scaling in linear transformations. Understanding how standard operations—such as scalar multiplication, inversion, transposition, and exponentiation—affect the determinant is crucial in various contexts within linear algebra.

Determinant of Scalar Multiplication

When a matrix is multiplied by a scalar, the determinant of the result is equal to the scalar raised to the power of the order (size) of the matrix multiplied by the determinant of the original matrix. This property is key to quickly evaluating how scaling transformation factors influence volume changes.

Determinant of the Inverse Matrix

For an invertible matrix, the determinant of its inverse is the reciprocal of the determinant of the matrix. This relationship is fundamental because it reflects the reversal of the transformation’s scaling effect and the importance of nonzero determinants in ensuring invertibility.

Determinant of the Transposed Matrix

Transposing a matrix does not change its determinant. This invariance is useful in computations combining transposition with other operations, and it shows that the determinant is independent of the order of multiplication in the context of row or column interchange.

Determinant of Matrix Powers

If a matrix is raised to a power, the determinant of the resulting matrix is equal to the determinant of the original matrix raised to that same power. This property simplifies the evaluation of the overall effect of repeated transformations.

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