# Linear Algebra Questions and Answers – Rank of Matrix in Row Echelon Form

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This set of Linear Algebra Questions and Answers for Freshers focuses on “Rank of Matrix in Row Echelon Form”.

1. Write Matrix corresponding to the following linear transformations.
y1=2x1-x2-x3
y2=3x3
y3=x1+x2
a) $$\begin{bmatrix}2&-1&-1\\0&0&3\\1&1&0\end{bmatrix}$$
b) $$\begin{bmatrix}2&-1&-1\\0&0&3\\1&0&0\end{bmatrix}$$
c) $$\begin{bmatrix}2&-1&-1\\0&0&3\\1&1&1\end{bmatrix}$$
d) $$\begin{bmatrix}2&-1&-1\\0&1&3\\1&1&0\end{bmatrix}$$

Explanation: In the given question,
We know that Linear Transformation is given by,
$$\begin{bmatrix}y1\\y2\\y3\end{bmatrix}\begin{bmatrix}2&-1&-1\\0&0&3\\1&1&0\end{bmatrix}\begin{bmatrix}x1\\x2\\x3\end{bmatrix}$$

Thus, the matrix for linear transformation is
$$\begin{bmatrix}2&-1&-1\\0&0&3\\1&1&0\end{bmatrix}$$.

2. Which of the following Linear Transformations is not correct for the given matrix?
$$\begin{bmatrix}1&2&-3\\-1&0&1\\2&1&0\end{bmatrix}$$
a) x1=1y1-3y2-3y3
b) x1=1y1-2y2-3y3
c) x2=-1y1+1y3
d) x3=2y1+y2

Explanation: In the given question,
X=$$\begin{bmatrix}1&2&-3\\-1&0&1\\2&1&0\end{bmatrix}$$Y
Thus,
x1=1y1-2y2-3y3
x2=-1y1+1y3
x3=2y1+y2.

3. For the linear transformation, X=$$\begin{bmatrix}2&1&1\\1&1&2\\1&0&-2\end{bmatrix}$$Y, find the Y co-ordinates for (1, 2, -1) in X.
a) (0, -2, 0)
b) (-1, 3, 1)
c) (-1, -2, 0)
d) (-1, 3, 0)

Explanation: In the given question,
X=(1, 2, -1)
$$\begin{bmatrix}1\\2\\-1\end{bmatrix}\begin{bmatrix}2&1&1\\1&1&2\\1&0&-2\end{bmatrix}\begin{bmatrix}y1\\y2\\y3\end{bmatrix}$$

y1– 2y3=-1
y2+4y3=3
y3=0
Thus Y (-1, 3, 0).