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 \(\displaystyle 0=det(A)\)