Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra and its Applications
Found in: Page 93
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

Answers without the blur.

Just sign up for free and you're in.


Short Answer

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1–4.

3. \[\left[ {\begin{array}{*{20}{c}}{\bf{0}}&I\\I&{\bf{0}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}W&X\\Y&Z\end{array}} \right]\]

The product is \[\left[ {\begin{array}{*{20}{c}}Y&Z\\W&X\end{array}} \right]\].

See the step by step solution

Step by Step Solution

Step 1: State the row-column rule

If the sum of the products of matching entries from row \(i\) of matrix A and column \(j\) of matrix B equals the item in row \(i\) and column \(j\) of AB, then it can be said that product AB is defined.

The product is shown below:

\({\left( {AB} \right)_{ij}} = {a_{i1}}{b_{1j}} + {a_{i2}}{b_{2j}} + ... + {a_{in}}{b_{nj}}\)

Step 2: Obtain the product

Compute the product by using the row-column rule, as shown below:

\(\begin{array}{c}\left[ {\begin{array}{*{20}{c}}0&I\\I&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}W&X\\Y&Z\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{0\left( W \right) + I\left( Y \right)}&{0\left( X \right) + I\left( Z \right)}\\{I\left( W \right) + 0\left( Y \right)}&{I\left( X \right) + 0\left( Z \right)}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{IY}&{IZ}\\{IW}&{IX}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}Y&Z\\W&X\end{array}} \right]\end{array}\)

Thus, \[\left[ {\begin{array}{*{20}{c}}0&I\\I&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}W&X\\Y&Z\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}Y&Z\\W&X\end{array}} \right]\].

Most popular questions for Math Textbooks


Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.