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) \(\begin{bmatrix}2&-1&-1\\0&1&3\\1&1&0\end{bmatrix}\)

View Answer

Answer: a
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
\(\begin{bmatrix}2&-1&-1\\0&0&3\\1&1&0\end{bmatrix}\).
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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
View Answer

Answer: a
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)
View Answer

Answer: d
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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn