A,B,C are 3 times 3 matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is det(A^TA^{-1})?1-93-19

Bevan Mcdonald 2021-02-21 Answered

A,B,C are \(3 \times 3\) matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is \(det(A^TA^{-1})\)?
1
-9
3
-1
9

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Expert Answer

curwyrm
Answered 2021-02-22 Author has 15973 answers

Step 1
Given that
A,B and C are \(3 \times 3\) matrices.
To find
\(det(A^TA^{-1})\)
Step 2
Given that:
A,B,C are \(3 \times 3\) matrices
det(A)=-3 , det(D)=-2, det(C)=6.
To find
\(det(A^TA^{-1})\)
We know that
\(det(AB)=detA \cdot detB\)
\(det(A^{-1})=\frac{1}{det(A)}\)
\(det(A^T)=det(A)\)
Hence consider
\(det(A^TA^{-1})\)
\(=det(A^T) \cdot det(A^{-1})\)
\(=det(A) \cdot \frac{1}{det(A)}\)
\(=(-3) \cdot \frac{1}{-3}\)
=1
\(\therefore det(A^TA^{-1})=1\)
option A

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