Determinant Properties

Let

A, B \in R^{3 \times 3}

. Suppose that

det (A) = 2

and

det (B) = 4

. What is

det (2 A^{T} B^{- 1})

The answer is 4.

Since $A$ and $B$ are $3 \times 3$ matrices, the product $A^{T} B^{- 1}$ is also $3 \times 3$ . Hence, $\begin{array}{rcl} det (2 A^{T} B^{- 1}) & = & 2^{3} det (A^{T} B^{- 1}) \\ = & 8 det (A^{T}) det (B^{- 1}) \\ = & 8 det (A) / det (B) \\ = & 2 \cdot 8 / 4 = 4. \end{array}$

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