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

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).

Sanfoundry Global Education & Learning Series – Linear Algebra.

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