Question

Let a be a matrix with three pivots. Select all correct statements.

1) The rank of matrix a is 3.
2) Matrix a is invertible.
3) Matrix a has a unique solution.
4) Matrix a is a square matrix.

Answer 1

The correct statements are: the rank of matrix a is 3, matrix a is invertible, and matrix a is a square matrix.

Step-by-step explanation:

The correct statements are:

1. The rank of matrix a is 3. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. Since a has three pivots, it means there are three linearly independent rows or columns. Therefore, the rank of matrix a is 3.
2. Matrix a is invertible. If a matrix has three pivots, it means it has a full rank. A matrix is invertible if and only if it has a full rank. Therefore, matrix a is invertible.
3. Matrix a is a square matrix. A matrix is square if it has the same number of rows and columns. Since a has three pivots, it means it has three rows and three columns, making it a square matrix.