2.4 The Inverse of a Matrix

For any two matrices A and B, if AB = In for some positive integer n, then A is invertible.

False

If A and B are invertible n × n matrices, then A + B is invertible.

False

If A is an invertible n × n matrix and the reduced row echelon form of [A B] is [In C], then C = B−1A.

False

If the reduced row echelon form of [A In] is [R B], then B = A−1.

False

A matrix is invertible if and only if its reduced row echelon form is an identity matrix.

True

A square matrix is invertible if and only if its reduced row echelon form has no zero row.

True

An n × n matrix is invertible if and only if its columns are linearly independent.

True

An n × n matrix is invertible if and only if its rows are linearly independent.

True

Any invertible matrix can be written as a product of elementary matrices.

True

For any two n × n matrices A and B, if AB = In, then BA = In .

True

For any two n × n matrices A and B, if AB = In, then A is invertible and A−1 = B.

True

If A is an n × n matrix such that Ax = b is consistent for every b in Rn, then Ax = b has a unique solution for every b in Rn .

True

If A is an n × n matrix such that the only solution of Ax = 0 is 0, then A is invertible.

True

If a square matrix has a column consisting of all zeros, then it is not invertible.

True

If a square matrix has a row consisting of all zeros, then it is not invertible.

True

If an n × n matrix has rank n, then it is invertible.

True

If an n × n matrix is invertible, then it has rank n.

True

If the reduced row echelon form of [A In] is [R B], then B is an invertible matrix.

True

If the reduced row echelon form of [A In] is [R B], then BA equals the reduced row echelon form of A.

True

Suppose that A is an invertible matrix and u is a solution of Ax = ⎡ ⎢⎢⎣ 5 6 7 8 ⎤ ⎥⎥⎦ . The solution of Ax = ⎡ ⎢⎢⎣ 5 6 9 8 ⎤ ⎥⎥⎦ differs from u by 2p3, where p3 is the third column of A−1.

True