Question

Consider two vectors$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$ and $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{w}}$in${\mathbf{ℝ}}^{\mathbf{3}}$. Form the matrix $\mathit{A}\mathbf{=}\left[\stackrel{\rightharpoonup }{v}x\stackrel{\rightharpoonup }{w}\stackrel{\rightharpoonup }{v}\stackrel{\rightharpoonup }{w}\right]$ . Express detA in terms of$||\stackrel{\rightharpoonup }{v}x\stackrel{\rightharpoonup }{w}||$. For which choices of $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$ and $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{w}}$ is Ainvertible?

Short answer

$det[\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}\stackrel{\rightharpoonup }{\mathrm{v}}\stackrel{\rightharpoonup }{\mathrm{w}}]={||\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}||}^2$

Therefore, the matrix is invertible if and only $\stackrel{\rightharpoonup }{v}$if $\stackrel{\rightharpoonup }{w}$and are non-parallel.

Step-by-step solution

Matrix Definition. 

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “mxn”

To find which choices of matrix is invertible. 

To solve,

$det[\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}\stackrel{\rightharpoonup }{\mathrm{v}}\stackrel{\rightharpoonup }{\mathrm{w}}]=\left[\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}},\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}\right]={||\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}||}^2$

For any given $\stackrel{\rightharpoonup }{\mathrm{v}},\stackrel{\rightharpoonup }{\mathrm{w}}\in {\mathrm{ℝ}}^3$, the vector $\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}$is perpendicular to both vectors, meaning it's never parallel to either one of them, so the given matrix is invertible if and only if $\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{and}\stackrel{\rightharpoonup }{\mathrm{w}}$are non-parallel.

Therefore, the matrix is invertible if and only if $\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{and}\stackrel{\rightharpoonup }{\mathrm{w}}$are non-parallel.

$det[\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}\stackrel{\rightharpoonup }{\mathrm{v}}\stackrel{\rightharpoonup }{\mathrm{w}}]={||\stackrel{\rightharpoonup }{\mathrm{v}}\mathrm{x}\stackrel{\rightharpoonup }{\mathrm{w}}||}^2$