Question

does this matrix have an inverse?

Answer 1

compute its determinant:$$\det\begin{bmatrix}3&6\\2&4\end{bmatrix}=3(4)-2(6)=12-12=0$$ergo it's not invertible

Answer 2

this is also obvious from the fact:$$\begin{bmatrix}3&6\\2&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=x\begin{bmatrix}3\\2\end{bmatrix}+y\begin{bmatrix}6\\4\end{bmatrix}=(x+2y)\begin{bmatrix}3\\2\end{bmatrix}$$in other words the image of the matrix is spanned by a single vector and is thus 1-dimensional whereas the domain is a 2-dimensional sapce

Answer 3

thank you!

Answer 4

"invertible" matrices must preserve the # of dimensions of a space... if the image (output) of the matrix is of lower dimension than the domain (input) then it is guaranteed that the transformation has "lost" information in a sense and is therefore not invertible

Answer 5

can you also help with this one?

Answer 6

it would be pointless for me to repeat the quesiton but:
http://www.mathsisfun.com/algebra/matrix-multiplying.html