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Row space of $$$\left[\begin{array}{ccc}1 & 2 & 3\\9 & 12 & 5\\5 & 7 & 4\end{array}\right]$$$

The calculator will find the row space of the $$$3$$$x$$$3$$$ matrix $$$\left[\begin{array}{ccc}1 & 2 & 3\\9 & 12 & 5\\5 & 7 & 4\end{array}\right]$$$, with steps shown.

Related calculator: Column Space Calculator

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Your Input

Find the row space of $$$\left[\begin{array}{ccc}1 & 2 & 3\\9 & 12 & 5\\5 & 7 & 4\end{array}\right]$$$.

Solution

The reduced row echelon form of the matrix is $$$\left[\begin{array}{ccc}1 & 0 & - \frac{13}{3}\\0 & 1 & \frac{11}{3}\\0 & 0 & 0\end{array}\right]$$$ (for steps, see rref calculator).

The row space is a space spanned by the nonzero rows of the reduced matrix.

Thus, the row space is $$$\left\{\left[\begin{array}{c}1\\0\\- \frac{13}{3}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{11}{3}\end{array}\right]\right\}$$$.

Answer

The row space of the matrix is $$$\left\{\left[\begin{array}{c}1\\0\\- \frac{13}{3}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{11}{3}\end{array}\right]\right\}$$$A.