Linear Algebra: 2.3 Invertible Matrices

¡Supera tus tareas y exámenes ahora con Quizwiz!

A =nxn is invertible if and only if...

A is row equivalent to I_n Any sequence of elementary row operations reducing A ~ I_n also transforms I_n ~ A^(-1)

If A is an invertible nxn matrix, then for each b in R^n...

Ax = b has the unique solution x = A^(-1)b

An nxn matrix A is said to be invertible if there is an nxn matrix C such that...

CA = I and AC = I

If C is 6x6 and the equation Cx=v is consistent for every v in R^6​, is it possible that for some v​, the equation Cx=v has more than one​ solution? Why or why​ not?

It is not possible. According to the Invertible Matrix Theorem that makes the matrix invertible. Since it is​ invertible, it has a unique solution.

The algorithm to find A^(-1) for a square matrix

Reduce the system [A I] ~ [I A^(-1)]

If an nxn matrix K cannot be row reduced to I_n​, what can you say about the columns of​ K? Why?

The columns of the matrix are linearly dependent and the columns do not span R^n.

Elementary matrix

a matrix obtained by performing a single elementary row operation on an identity matrix

The determinant of A is written as...

det A = ad - bc

The product of nxn invertible matrices is...

invertible

If ad - bc = 0 then A is...

not invertible

The inverse of AB is the product of the inverses of A and B in ________ order.

reverse

The inverse is _______ and denoted ______.

unique, A^(-1)


Conjuntos de estudio relacionados

Astronomy Today: Stars and Galaxies: Chapter 20

View Set

Lahey Psychology Exam 1 and 20 other sets

View Set

Health: Policy Provisions, Clauses, and Riders (2:3)

View Set

Federal, Confederal, and Unitary Governments

View Set

Lesson 9. Words Relating to Cheapness or Care with Spending Money (austere, avaricious, frugal, mercenary, miserly, parsimonious, penurious, thrifty)

View Set

Transport of Oxygen in Body Fluids

View Set

The Pathway of Blood Flow Through the Heart (The Coronary Circulation)

View Set