9 ible matrix, prove that 5A is an invertible matrix.[³].= b₁,e sameis the(a) byb4].and DIn Exercises 9 and 10, mark each statement True or False. Justifyeach answer.9. a In order for a matrix B to be the inverse of A, bothequations AB = I and BA = I must be true.b. If A and B are n x n and invertible, then ATB¹ is theinverse of AB.a- [ºc. If A =e.10. /a.blik 8X1101+10bdand ab-cd #0, then A is invertible.d. If A is an invertible n x n matrix, then the equationAx=b is consistent for each b in R".Each elementary matrix is invertible.A product of invertible n x n matrices is invertible, andthe inverse of the product is the product of their inversesin the same order.c. If A =b. If A is invertible, then the inverse of AT is A itself.= [a b]dand ad bc, then A is not invertible.=d. If A can be row reduced to the identity matrix, then A mustbe invertible. DMod2.20e. If A is invertible, then elementary row operations thatreduce A to the identity I, also reduce A-¹ to In.11. Let A be an invertible n x n matrix, and let B be an n x pmatrix. Show that the equation AX = B has a unique solu-tion AB.12. Let A be an invertible n x n matrix, and let B be an n x p ma-trix. Explain why A¹B can be computed by row reduction:

each answer. 9. a In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true. b. If A and B are n x n and invertible, then ATB¹ is the inverse of AB. b - [a 2] d d. If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R". e. Each elementary matrix is invertible. c. If A = and and ab-cd0, then A is invertible. is invertible and

each answer. 9. a In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true. b. If A and B are n x n and invertible, then ATB¹ is the inverse of AB. b - [a 2] d d. If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R". e. Each elementary matrix is invertible. c. If A = and and ab-cd0, then A is invertible. is invertible and

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Matrix Multiplication

A matrix is an entity composed of components arranged in rows and columns. The rows and columns of a matrix determine the dimension of a matrix. A matrix containing 3 rows and 4 columns is of dimension 3x4. Dimensions in matrix arithmetic are very important since some operations are not possible unless matrices have identical dimensions.

Question

ible matrix, prove that 5A is an invertible matrix.
[³].
= b₁,
e same
is the
(a) by
b4].
and D
In Exercises 9 and 10, mark each statement True or False. Justify
each answer.
9. a In order for a matrix B to be the inverse of A, both
equations AB = I and BA = I must be true.
b. If A and B are n x n and invertible, then ATB¹ is the
inverse of AB.
a
- [º
c. If A =
e.
10. /a.
blik 8X1101
+10
b
d
and ab-cd #0, then A is invertible.
d. If A is an invertible n x n matrix, then the equation
Ax=b is consistent for each b in R".
Each elementary matrix is invertible.
A product of invertible n x n matrices is invertible, and
the inverse of the product is the product of their inverses
in the same order.
c. If A =
b. If A is invertible, then the inverse of AT is A itself.
= [a b]
d
and ad bc, then A is not invertible.
=
d. If A can be row reduced to the identity matrix, then A must
be invertible. D
Mod2.20
e. If A is invertible, then elementary row operations that
reduce A to the identity I, also reduce A-¹ to In.
11. Let A be an invertible n x n matrix, and let B be an n x p
matrix. Show that the equation AX = B has a unique solu-
tion AB.
12. Let A be an invertible n x n matrix, and let B be an n x p ma-
trix. Explain why A¹B can be computed by row reduction:

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