Which of the following statements characterizes a non-singular matrix?

Which of the following statements characterizes a non-singular matrix?

• A. A matrix with determinant zero
• B. A matrix that has an inverse
• C. A matrix with all zero entries
• D. A matrix with no inverse

Answer: (B)

Non-singular means the matrix is invertible. That happens exactly when you can find a matrix that serves as its inverse, so you can always solve Ax = b uniquely for any b. Equivalently, the determinant is not zero and the columns (or rows) are linearly independent. The statement that best captures this is that the matrix has an inverse. If the determinant were zero, or if there were no inverse, that would describe a singular (non-invertible) matrix. The option describing all-zero entries also fails because that matrix has determinant zero and no inverse.

Non-singular means the matrix is invertible. That happens exactly when you can find a matrix that serves as its inverse, so you can always solve Ax = b uniquely for any b. Equivalently, the determinant is not zero and the columns (or rows) are linearly independent.

The statement that best captures this is that the matrix has an inverse. If the determinant were zero, or if there were no inverse, that would describe a singular (non-invertible) matrix. The option describing all-zero entries also fails because that matrix has determinant zero and no inverse.