M-R- 2and C =N: ind the fllowing:-D -R]Given A =„find the following:12Rа. АСb. 2B + сТc. A-1 -1

Given information: A=-D3NK B=M0L-22R C=M-D-RL06

Answered: M -R- 2and C =N: ind the fllowing: -D…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 73CR

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Question

M
-R- 2and C =N: ind the fllowing:
-D -R]
Given A =
„find the following:
12
R
а. АС
b. 2B + сТ
c. A-1 -1

Expert Solution

Step 1

Given information:

$A = (\begin{matrix} - D & 3 \\ N & K \end{matrix})$

$B = (\begin{matrix} M & 0 \\ L & - 2 \\ 2 & R \end{matrix})$

$C = (\begin{matrix} M & - D & - R \\ L & 0 & 6 \end{matrix})$

Step 2

a. To Find AC multiply the rows of the first matrix by the columns of the second matrix.

$\begin{array}{rcl} A C & = & (\begin{matrix} - D & 3 \\ N & K \end{matrix}) (\begin{matrix} M & - D & - R \\ L & 0 & 6 \end{matrix}) \\ = & (\begin{matrix} (- D) M + 3 L & (- D) (- D) + 3 \cdot 0 & (- D) (- R) + 3 \cdot 6 \\ N M + K L & N (- D) + K \cdot 0 & N (- R) + K \cdot 6 \end{matrix}) \\ = & (\begin{matrix} - D M + 3 L & D^{2} & D R + 18 \\ N M + K L & - D N & - N R + 6 K \end{matrix}) \end{array}$

Step 3

b. To find $2 B + C^{T}$ first, find $C^{T}$ and 2B

$\begin{array}{rcl} C^{T} & = & {(\begin{matrix} M & - D & - R \\ L & 0 & 6 \end{matrix})}^{T} \\ = & (\begin{matrix} M & L \\ - D & 0 \\ - R & 6 \end{matrix}) \end{array}$

$\begin{array}{rcl} 2 B & = & 2 (\begin{matrix} M & 0 \\ L & - 2 \\ 2 & R \end{matrix}) \\ = & (\begin{matrix} 2 M & 2 \cdot 0 \\ 2 L & 2 (- 2) \\ 2 \cdot 2 & 2 R \end{matrix}) \\ = & (\begin{matrix} 2 M & 0 \\ 2 L & - 4 \\ 4 & 2 R \end{matrix}) \end{array}$

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M -R- 2and C =N: ind the fllowing: -D -R] Given A = „find the following: 12 R а. АС b. 2B + сТ c. A-1 -1