Question

Let  be an invertible matrix. Select which of the following statements must be true:

1.A² is invertible

2. Ax=B has a unique solution for any b

3. A can have more rows than columns

4.det(A) =0

Answer 1

Matrix A is invertible if and only if it is a square matrix and $\det A\not=0.$

1.A² is invertible.

 True.

$$\det(A^2)=\det A(\det A)\not=0$$

2. Ax=B has a unique solution for any b.

True.

If A is an n × n invertible matrix, then the system of linear equations given by Ax = B has the unique solution $x = A^{−1}B.$

3. A can have more rows than columns.

False.

If A is invertible then A is a square matrix.

4.det(A) =0

False.

If matrix A is invertible then $\det A\not=0.$