2.2-3.1

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

False. (AB)^-1=B^-1A^-1.

If A=[a b;c d] and ab-cd =/=0, then A is invertible.

False. A is invertible is ad-bc =/=0.

A product of invertible n x n matrices is invertible, and the inverse of the product of their matrices in the same order.

False. It is invertible, but the inverses in the product of the inverses in the reverse order.

The determinant of a triangular matrix is the sum of the entries of the main diagonal

False. It is the product of the diagonal entries.

If the linear transformation x->Ax maps Rn into Rn then A has n pivot points.

False. Since A is n x n the linear transformation maps Rn into Rn. This doesn't tell us anything about A.

The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column.

False. The cofactor is the determinant of this Aij times -1^(i+j)

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

False. They also reduce the identity to A-1

The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.

False. We can expand down any row or column and get same determinant

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. We need to know more about A like if it is invertible

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

TRUE

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

TRUE. We'll see later that for square matrices AB=I then there is some C such that BC=I

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

True

If Atrans is not invertible, then A is not invertible

True

If the columns of A are linearly independent, then the columns of A span Rn

True

If the columns of A span Rn, then the columns are linearly independent

True

If the equation Ax=0 has only the trivial solution, the A is row equivalent to the nxn identity matrix

True

If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix

True

If the equation Ax=b has at least one solution for each b in Rn, then the solution is unique for each b

True

If there 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

An n x n determinant is defined by determinants of (n-1) x (n-1) submatrices

True.

If A=[a b;c d], and ad=bc, then A is not invertible

True. A is invertible if ad-bc =/=0 but if ad=bc then ad-bc=0

Each elementary matrix is invertible.

True. Let K be the elementary row operation required to change the elementary matrix back into the identity. If we preform K on the identity, we get the inverse.

If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn

True. Since A is invertible we have that x= A^-1b

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

True. The algorithm presented in this chapter tells us how to find the reverse in this case.

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

True. This comes from the "all false" part of Thm 8.