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I have a matrix like this: enter image description here

I search for mathmatical operation that would turn a given column (say $T_3$ or $T_4$) into all zeros not changing anything else about this matrix. How to do such thing in math?

DuckQueen
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  • In math, you could just say, "Let $A$ be the previous matrix, but with column 3 replaced by a zero vector." If that's not what you're looking for you'll have to be more precise by your "how to do such a thing". – Mees de Vries Jul 23 '18 at 09:03
  • Consider multiplying on the right by a square matrix of the appropriate size whose entries are the same as the identity except the given column is all $0$s. This is the elementary matrix corresponding to the column operation of multiplying a column by $0$. – Osama Ghani Jul 23 '18 at 09:04

1 Answers1

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Recall that in general for $A_{m\times n}B_{n\times n}=C_{m\times n}$ the i-th column of $C$ is the combination of the columns of $A$ with respect to the coefficient of the i-th column of $B$.

Then we can multiply by the matrix

\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}

to obtain a matrix with $T_4=0$ and similarly we can obtain a matrix with $T_3=0$.

user
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  • Will it keep $T_2$ values if they are not on the diagonal line? – DuckQueen Jul 23 '18 at 09:09
  • @DuckQueen Yes of course, let try by direct check. Note that $A_{5x4}I_{4x4}=A_{5x4}$ and also recall that in general for $AB=C$ the i-th column of $C$ is the combination of the columns of $A$ with respect to the coefficient of the i-th column of $B$. – user Jul 23 '18 at 09:12
  • And how shell I do it when my matrix is not square (20 cols. 30 rows)? – DuckQueen Jul 23 '18 at 09:19
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    @DuckQueen Also in the case of the OP it is not square, in general we have $A_{mxn}I_{nxn}=A_{mxn}$ for any $m$ and $n$. – user Jul 23 '18 at 10:10
  • Thank you so much!))) – DuckQueen Jul 23 '18 at 10:19