Linear Algebra concepts 2.2&2.3

If A is invertible, then elementary row operations that reduce A to identity In also reduce A^-1 to ln

False

If the linear transformation (x) --> Ax maps R^n into r^n, has n pivot positions

False

If A and B are n x n and invertible, then A^-1 B^-1 is the inverse of AB

False (B^-1 A^-1 is the inverse of AB)

If A is an n x n matrix then the equation Ax=b has at least one solution for each b in R^n

False (By the invertible matrix theorem Ax=b has at least one solution for each b in R^n ONLY if a matrix (matrix A) 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

False (If A and B are invertible matrices. then (AB)^-1 =B^-1A^-1)

If a= matrix [abcd] and ab-cd is not equal to 0, then A is not? invertible

False (If ad-bc is not equal to 0, then A is invertible)

In order for a matrix B to be the inverse of​ A, both equations AB=I and BA = I must be true

True

If A can be row reduced to the identity matrix, then A must be invertible

True ( A can be row reduced to the identity​ matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is​ invertible, A is invertible.)

If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions

True ( By the invertible matrix theorem if the equation Ax=0 has a nontrivial solution, then the matrix A is not invertible. Therefore, A has fewer than n pivot positions)

If A is invertible, then the inverse of A^-1 is A itself

True ( Since A^-1 is the inverse of A, A^-1 A=Identity matrix=AA^-1, A is the inverse of A^-1)

If the columns of A are linearly independent, then the columns of A span R^n

True ( by the Invertible Matrix Theorem if the columns of A are linearly​ independent, then the columns of A must span R^n)

If the equation Ax=o only has the trivial solution, then A is row equivalent to n x n identity matrix

True ( by the invertible matrix theorem if the equation Ax=0 has only the trivial solution, then the matrix is invertible )

If the is an n x n matrix D such that AD=I, then there is also an n x n matrix C such that CA=I.

True ( by the invertible matrix theorem if there is an n x n matrix such that AD=I THEN IT MUST BE TRUE THAT THERE IS ALSO AN N X N matrix C such that CA=I )

If A=[abcd] and ad=bc, then A is not invertible

True ( if ad=bc then ad-bc=0)

If there is a b in R^n such that the equation Ax=b is inconsistent, then the transformation x--> Ax is not one to one

True (If there is a b in R^n such that the equation Ax=b is inconsistent, then the equation Ax=b does not have atleast one solution for each b in R^n and this makes A not invertible)

If A is an invertible n x n matrix, then the equation Ax=b is consistent for each b in R^n

True (Since A is invertible, A^-1 b exists for all b in R^n)

If the equation Ax=b has at least one solution for each b in R^n then the solution is unique for b

True (by the invertible matrix theorem if Ax=b has atleast one solution for each b in R^n, then the matrix A is invertible, then according the to invertible matrix theorem the solution is unique for each b)

Each elementary matrix is invertible.

True (since each elementary matrix corresponds to a row​ operation, and every row operation is​ reversible, every elementary matrix has an inverse matrix.)

If the columns of A span R^n, then the columns are linearly independent

True (the invertible matrix theorem states that if the columns of A span R^n, then the matrix A is invertible, therefore, the columns are linearly independent)