91Ó°ÊÓ

Find the determinants in Exercises \(15-20,\) where $$ \left|\begin{array}{ccc}{a} & {b} & {c} \\ {d} & {e} & {f} \\ {g} & {h} & {i}\end{array}\right|=7 $$ $$ \left[\begin{array}{ccc}{a+d} & {b+e} & {c+f} \\ {d} & {e} & {f} \\ {g} & {h} & {i}\end{array}\right] $$

The expansion of a \(3 \times 3\) determinant can be remembered by the following device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals: Graph cannot copy Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises \(15-18 .\) Warning: This trick does not generalize in any reasonable way to \(4 \times 4\) or larger matrices. $$ \left|\begin{array}{rrr}{2} & {-3} & {3} \\ {3} & {2} & {2} \\ {1} & {3} & {-1}\end{array}\right| $$

Let \(S\) be the parallelogram determined by the vectors $$\mathbf{b}_{1}=\left[\begin{array}{r}{-2} \\ {3}\end{array}\right] \text { and } \mathbf{b}_{2}=\left[\begin{array}{r}{-2} \\ {5}\end{array}\right], \text { and let } A=\left[\begin{array}{rr}{6} & {-3} \\ {-3} & {2}\end{array}\right]$$ Compute the area of the image of \(S\) under the mapping \(\mathbf{x} \mapsto A \mathbf{x} .\)

Let \(A\) and \(B\) be \(4 \times 4\) matrices, with det \(A=-3\) and \(\operatorname{det} B=-1\) . Compute: \(\begin{array}{lll}{\text { a. } \operatorname{det} A B} & {\text { b. } \operatorname{det} B^{5}} & {\text { c. } \operatorname{det} 2 A} \\ {\text { d. } \operatorname{det} A^{T} B A} & {\text { e. } \operatorname{det} B^{-1} A B} & {}\end{array}\)

[M] Construct a random \(4 \times 4\) matrix \(A\) with integer entries between \(-9\) and \(9 .\) How is det \(A^{-1}\) related to det \(A ?\) Experiment with random \(n \times n\) integer matrices for \(n=4\) 5, and \(6,\) and make a conjecture. Note: In the unlikely event that you encounter a matrix with a zero determinant, reduce it to echelon form and discuss what you find.