I am getting an error, While I am trying to connect (LocalDB)\MSSQLLocalDB through SQL Server management studio. I also tried to login with default database as master the error is same.
Here is the Server details.
The following command through sqllocaldb utility works for me.
sqllocaldb stop mssqllocaldb
sqllocaldb delete mssqllocaldb
sqllocaldb start "MSSQLLocalDB"
After that I restarted the sql server management studio, and it is successfully established connection through (LocalDB)\MSSQLLocalDB
Right click your server and go to Properties, the select the Security Under Server Authetication section, You will find that “Windows Authetication Mode” radio button selected. So, your work is to select “Sql Server and Windows Authetication Mode” radio button. Right click your server again and select restart. Then follow below steps. Try follwing steps:
(a)Expand the SQL Server
(c) Expand Security node
(d) Select the Particular Login (User)
(e) Press right button and select properties of the login
(f) Go to Status table Enble the User
来源：https://stackoverflow.com/questions/36950078/cannot-connect-to-localdb-mssqllocaldb-login-failed-for-user-user-pc-user