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Question: Is there a way to increase only one row/column in a matrix with a scalar?

We learned about multiplying matrices recently, and I was wondering how you would use a scalar to multiply against a matrix to increase only one of its rows or columns.

For example, I know (from messing around) that

$$\begin{bmatrix}1.5 & 0 &0\\0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix} \times \begin{bmatrix}16.5 & 18 &17.5\\12.5 & 14 & 17\\ 16 & 19.5 & 18\end{bmatrix}$$

will increase the first column. But how would you figure that out by just looking at the matrix? I think it has something to do with the identity element...

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  • $\begingroup$ Your example operates on the first row. To operate on columns, you have to multiply on the rught. $\endgroup$ – Bernard Oct 31 '16 at 15:41
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You want this:

$$\begin{bmatrix}a & 0 &0\\0 & b & 0\\ 0 & 0 & c\end{bmatrix}$$

It will multiply the first column by $a$, the second column by $b$ and the third column by $c$.

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To multiply the $i$th row by $\alpha$ and keep the rest unchanged, multiply to the left by the matrix obtained from the identity by replacing the $1$ at the $(i,i)$th position with $\alpha$.

To multiply the $i$th column by $\alpha$ and keep the rest unchanged, multiply with the same matrix as above but from the right.

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