91Ó°ÊÓ

Use the Principle of Mathematical Induction to prove that the determinant of the \(n \times n\) identity matrix \(I_{n}\) is equal to 1 .

Some of the statements are false, and some are conjectures that have defied all previous attempts at proof or disproof.\(\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}\)

Show that the inverse of an orientation-preserving linear map is orientationpreserving. What about the inverse of an orientation-reversing linear map?

Show that det \(\left[\begin{array}{llr}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]=\operatorname{det}\left[\begin{array}{llr}a & d & g \\ b & e & h \\ c & f & i\end{array}\right]\).

Use Cramer's rule to solve the system $$ \begin{aligned} &7 x+3 y=2 \\ &2 x-4 y=-3 \end{aligned} $$

Some of the statements are false, and some are conjectures that have defied all previous attempts at proof or disproof.\(\sum_{k=1}^{n} k^{3}=\frac{n^{2}(n+1)^{2}}{4}\)

When the normal vector for a plane is computed by taking the cross product of a pair of direction vectors for the plane, what assures us that the normal vector will be nonzero?

Prove Theorem 7.16, part b: \(\mathbf{v} \times(\mathbf{w}+\mathbf{x})=(\mathbf{v} \times \mathbf{w})+(\mathbf{v} \times \mathbf{x})\).

Use Cramer's rule to solve the system $$ \begin{array}{r} 3 x-2 y+2 z=1 \\ x+4 z=2 \\ 2 x+y-z=0 \end{array} $$

Prove that the determinant of the \(n \times n\) zero matrix is equal to 0 .