a) SHOW that a matrix with a row of zeros cannot have an inverse
b) SHOW that a matrix with a column of zeros cannot have an inverse
how do i show this? i have no clue
help would be much appreciated
A matrix is invertible iff. the $\displaystyle det\neq0$. What happens if a column of row is alls zeros?
Definition:
The determinant of an nxn matrix A, denoted det(A), is a scalar associated with the matrix A that is defined inductively as
$\displaystyle det(A)=
\begin{cases}
a_{11}, & \mbox{if }n=1 \\
a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1
\end{cases}
$ where $\displaystyle A_{1j}=(-1)^{1+j}det(M_{1j}),\ j=1,...,n$ are the cofactors associated with the entries in the first row of A.
Leon, S. (2010). Linear algebra with applications. Upper Saddle River, NJ: Pearson.
a.
By expanding across the row of all zeros, each term of the cofactor expansion will have a factor of 0. Hence, the sum will equal $\displaystyle 0=det(A)$
$\displaystyle det(A)=
\begin{cases}
a_{11}, & \mbox{if }n=1 \\
a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1
\end{cases} $
$\displaystyle det(A)=
\begin{cases}
a_{11}, & \mbox{if }n=1 \\
a_{11}A_{11}+a_{21}A_{21}+\dots+a_{n1}A_{n1}, & \mbox{if }n>1
\end{cases} $
If we are expanding along the rows (first def) or the columns (second def), what are the values of $\displaystyle a_{ij}$?
Or how about this if you can do it for columns.
$\displaystyle det(A)=det(A^T)$