15. B: If
is non-zero, then
. We have
![\det B=4\cdot0-1\cdot2=-2](https://tex.z-dn.net/?f=%5Cdet%20B%3D4%5Ccdot0-1%5Ccdot2%3D-2)
so
![\det B^{-1}=-\dfrac12](https://tex.z-dn.net/?f=%5Cdet%20B%5E%7B-1%7D%3D-%5Cdfrac12)
16. C: In general,
for two matrices
, but equality holds if
. D is incorrect because
.
17. C: This follows from a property of the determinant:
![\det(kB)=k^n\det B](https://tex.z-dn.net/?f=%5Cdet%28kB%29%3Dk%5En%5Cdet%20B)
where
is the size of the (square) matrix
(and "size" refers to the number of rows or columns, both of which are the same).
18. A: A matrix has no inverse if its determinant is 0. The determinant of (A) is 0 because it contains a row made up entirely of 0s.
19. B: The matrix product
only exists if the number of columns of
is equal to the number of rows of
. In (a), the first matrix has 1 column and the other has 2 rows, so multiplication is invalid. In (c), the product on the left side would produce
![\begin{bmatrix}1&-3\\-5&1\end{bmatrix}\begin{bmatrix}-3\\14\end{bmatrix}=\begin{bmatrix}-45\\29\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D1%26-3%5C%5C-5%261%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D-3%5C%5C14%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D-45%5C%5C29%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7Dx%5C%5Cy%5Cend%7Bbmatrix%7D)
so that
and
, but these values don't work with the given equations because -45 - 3(29) = -132, not -3. In (d), the product on the left side is
![\begin{bmatrix}-3\\14\end{bmatrix}\begin{bmatrix}x&y\end{bmatrix}=\begin{bmatrix}-3x&-3y\\14x&14y\end{bmatrix}=\begin{bmatrix}1&-3\\-5&1\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D-3%5C%5C14%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7Dx%26y%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D-3x%26-3y%5C%5C14x%2614y%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D1%26-3%5C%5C-5%261%5Cend%7Bbmatrix%7D)
but then this would mean both
and
, which are not consistent and have no solution.