Matricies Proof

Apr 2010
30
0
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
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
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 \(\displaystyle 0=det(A)\)
 
Last edited:
Apr 2010
30
0
so if a matrix consists of a column of zeros, it's \(\displaystyle det=0\). do i prove it by solving for the determinant?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
so if a matrix consists of a column of zeros, it's \(\displaystyle det=0\). do i prove it by solving for the determinant?
That is what I would do, because when you solve the det of a matrix, you expand down the easiest row (usually the row with the most zeros).
 
Apr 2010
30
0
if i was to do it with any 3x3 matrix, i use the expansion of minors method.
i can seem to do it for a column of zeros. how is it done for a row?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
if i was to do it with any 3x3 matrix, i use the expansion of minors method.
i can seem to do it for a column of zeros. how is it done for a row?
\(\displaystyle det(A)=
\begin{cases}
a_{11}, & \mbox{if }n=1 \\
a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1
\end{cases} \)

\(\displaystyle det(A)=
\begin{cases}
a_{11}, & \mbox{if }n=1 \\
a_{11}A_{11}+a_{21}A_{21}+\dots+a_{n1}A_{n1}, & \mbox{if }n>1
\end{cases} \)

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.

\(\displaystyle det(A)=det(A^T)\)
 
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