Which statement correctly describes a matrix that is proportional to the identity matrix?

Which statement correctly describes a matrix that is proportional to the identity matrix?

• A. All diagonal elements are equal and all off-diagonal elements are zero.
• B. All diagonal elements are zero.
• C. All off-diagonal elements are equal to the diagonal elements.
• D. The matrix has nonzero off-diagonal elements.

Answer: (A)

A matrix that is proportional to the identity matrix is a scalar multiple of I, meaning A = cI for some constant c. In such a matrix, every diagonal entry equals the same value c, and every off-diagonal entry is zero. That’s why the statement describing a matrix proportional to the identity is that all diagonal elements are equal and all off-diagonal elements are zero. For example, multiplying the identity by 5 gives a matrix with 5s on the diagonal and zeros elsewhere. The other descriptions don’t match this pattern. If the diagonal were zero, off-diagonal entries could be nonzero, which would not be a scalar multiple of the identity unless the whole matrix were just zero. If every off-diagonal entry equals the diagonal entry, you’d have a matrix with all entries equal, not a scalar multiple of the identity (except in the trivial zero case). Nonzero off-diagonal entries directly contradict being proportional to the identity.

A matrix that is proportional to the identity matrix is a scalar multiple of I, meaning A = cI for some constant c. In such a matrix, every diagonal entry equals the same value c, and every off-diagonal entry is zero. That’s why the statement describing a matrix proportional to the identity is that all diagonal elements are equal and all off-diagonal elements are zero. For example, multiplying the identity by 5 gives a matrix with 5s on the diagonal and zeros elsewhere.

The other descriptions don’t match this pattern. If the diagonal were zero, off-diagonal entries could be nonzero, which would not be a scalar multiple of the identity unless the whole matrix were just zero. If every off-diagonal entry equals the diagonal entry, you’d have a matrix with all entries equal, not a scalar multiple of the identity (except in the trivial zero case). Nonzero off-diagonal entries directly contradict being proportional to the identity.