Matrices Questions and Answers – Solving Equations by Crout’s Method

«
»

This set of Matrices Multiple Choice Questions & Answers (MCQs) focuses on “Solving Equations by Crout’s Method”.

1. Solve the given equations using Crout’s Method to get value of z.

2x + 3y + z = -1
5x + y + z = 9
3x + 2y + 4z = 11
advertisement

a) \(\frac{22}{7}\)
b) 8
c) \(\frac{21}{8}\)
d) \(\frac{32}{7}\)
View Answer

Answer: c
Explanation: For the given sets of equations,
The Matrix form is given by
\(\begin{bmatrix}2&3&1\\5&1&1\\3&2&4\end{bmatrix}
\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}-1\\9\\11\end{bmatrix}\)
As the form of AX=B
Let A be assumed to be LU
\(\begin{bmatrix}2&3&1\\5&1&1\\3&2&4\end{bmatrix}=\begin{bmatrix}a&0&0\\b&c&0\\d&e&f\end{bmatrix}
\begin{bmatrix}1&g&h\\0&1&i\\0&0&1\end{bmatrix}=\begin{bmatrix}a&ag&ah\\b&bg+c&bh+ci\\d&dg+e&dh+ei+f\end{bmatrix}\)
By comparing both sides
a=2, b=5, d=3
g=\(\frac{3}{2}\), c=\(\frac{-13}{2}\), e=\(\frac{-5}{2}\), h=\(\frac{1}{2}\), f=\(\frac{40}{13}\) and i=\(\frac{3}{13}\)
Thus,
L=\(\begin{bmatrix}2&0&0\\5&\frac{-13}{2}&0\\3&\frac{-5}{2}&\frac{40}{13}\end{bmatrix}\) and U=\(\begin{bmatrix}1& \frac{3}{2}&\frac{1}{2}\\0&1&\frac{3}{13}\\0&0&1\end{bmatrix}\)
Now LY=B where Y=UX
\(\begin{bmatrix}2&0&0\\5&\frac{-13}{2}&0\\3&\frac{-5}{2}&\frac{40}{13}\end{bmatrix}
\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}-1\\9\\11\end{bmatrix}\)
Comparing Directly,
y1=\(\frac{-1}{2}\) y2=\(\frac{-23}{13}\) y3=\(\frac{21}{8}\)
Assume UX=Y
\(\begin{bmatrix}1& \frac{3}{2}&\frac{1}{2}\\0&1&\frac{3}{13}\\0&0&1\end{bmatrix}
\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}\frac{-1}{2}\\\frac{-23}{12}\\\frac{21}{8}\end{bmatrix}\)
Comparing both sides we get.
Z=\(\frac{21}{8}\)
Thus, the value of y is \(\frac{21}{8}\).

Sanfoundry Global Education & Learning Series – Matrices.

To practice all areas of Matrices, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
advertisement
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