Which statement describes the form of the identity matrix?

Which statement describes the form of the identity matrix?

• A. Diagonal of 1s from the main diagonal with zeros elsewhere
• B. Diagonal of 0s with ones elsewhere
• C. All entries equal to 1
• D. 1s on the diagonal and some off-diagonal zeros

Answer: (A)

The main idea is that the identity matrix acts as the multiplicative identity in matrix multiplication, so it must leave any compatible matrix unchanged when multiplied. To do that, every position on the main diagonal must be 1, and every off-diagonal position must be 0. This gives the standard form with 1s along the main diagonal and 0s elsewhere, sometimes written as I_n. For example, the 2×2 case is [[1, 0], [0, 1]]. Any deviation, such as placing 1s off the diagonal, would alter the product and not act as the identity. A matrix with all entries equal to 1 would not preserve a vector, and saying there are “some” off-diagonal zeros doesn’t guarantee that every off-diagonal entry is zero, which is required for the identity form.

The main idea is that the identity matrix acts as the multiplicative identity in matrix multiplication, so it must leave any compatible matrix unchanged when multiplied. To do that, every position on the main diagonal must be 1, and every off-diagonal position must be 0. This gives the standard form with 1s along the main diagonal and 0s elsewhere, sometimes written as I_n.

For example, the 2×2 case is [[1, 0], [0, 1]]. Any deviation, such as placing 1s off the diagonal, would alter the product and not act as the identity. A matrix with all entries equal to 1 would not preserve a vector, and saying there are “some” off-diagonal zeros doesn’t guarantee that every off-diagonal entry is zero, which is required for the identity form.