Answer to Question #137167 in Linear Algebra for Lethu

Question #137167

Suppose A is a 3x3 matrix such that det(A) =16.Find the value of det[(4(A^-1)^T]


1
Expert's answer
2020-10-07T17:44:52-0400

Taking into account that det(M1)=1det(M)\det(M^{-1})=\frac{1}{\det (M)} and det(MT)=det(M)\det(M^T)=\det (M) for any matrix MM, we conclude that det((A1)T)=det(A1)=1detA=116.\det ((A^{-1})^T)=\det (A^{-1})=\frac{1}{\det A}=\frac{1}{16}.

Since det(kM)=k3det(M)\det(kM)=k^3\det(M) for any 3×33\times 3 matrix MM,

det(4(A1)T)=43det((A1)T)=43116=4.\det (4(A^{-1})^T)=4^3\det ((A^{-1})^T)=4^3\frac{1}{16}=4.


Answer: 4.4.

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