Question

If a matrix A can be row reduced to the identity matrix, then A must be ________.

1) invertible
2) non-invertible
3) symmetric
4) orthogonal

Answer 1

If matrix A can be row reduced to the identity matrix, A must be invertible, which means there exists a unique inverse matrix A⁻¹ such that A multiplied by A⁻¹ equals the identity matrix.

Step-by-step explanation:

If a matrix A can be row reduced to the identity matrix, then A must be invertible. An invertible matrix, also known as a nonsingular or nondegenerate matrix, is a square matrix that can be transformed into the identity matrix I through a finite series of elementary row operations.

The process of row reducing a matrix to the identity matrix essentially finds a sequence of operations that transforms the original matrix A into I, and the existence of such operations implies that there is a unique inverse matrix A⁻¹ that, when multiplied by A, yields the identity matrix.