Q16E Use the given partitions to comp... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q16E (page 85)

Use the given partitions to compute the products in Exercises 15 and 16. Check your work by computing the same products without using a partition. Show all your work.
16. $[\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 4 & 5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$

Short Answer

Expert verified

The product is $[\begin{matrix} 1 & 2 & 3 & 5 \\ 3 & 4 & 7 & 9 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$ .

Step by step solution

Block matrix

Let the given matrix . $A = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], B = [\begin{matrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 4 & 5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$

In the blocking format given matrix can be written as

$A = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}], B = [\begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix}],$

Multiplication of block matrix

Multiplication of block matrix is done like as follows.

$\begin{array}{l} A B = [\begin{matrix} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] & [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] & [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] \end{matrix}] \\ A B = [\begin{matrix} \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} & [\begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix}] + [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] \\ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} & \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \end{matrix}] \\ A B = [\begin{matrix} 1 & 2 & 3 & 5 \\ 3 & 4 & 7 & 9 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}] \end{array}$

Checking the product

Here, we will check the product by doing simple multiplication.

$A B = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 4 & 5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}] = [\begin{matrix} 1 & 2 & 3 & 5 \\ 3 & 4 & 7 & 9 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$

Hence, the product is $[\begin{matrix} 1 & 2 & 3 & 5 \\ 3 & 4 & 7 & 9 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$ .

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91影视

Short Answer

Step by step solution

Block matrix

Multiplication of block matrix

Checking the product

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91影视

Use the given partitions to compute the products in Exercises 15 and 16. Check your work by computing the same products without using a partition. Show all your work.16.[1010010100100001]1223344500120034

Short Answer

Step by step solution

Block matrix

Multiplication of block matrix

Checking the product

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.