Identity matrix is defined as which of the following?

Identity matrix is defined as which of the following?

• A. A square matrix with ones on the diagonal and zeros elsewhere
• B. A matrix with all elements equal to one
• C. A diagonal matrix with arbitrary diagonal values
• D. A matrix with zeros on the diagonal and ones off diagonal

Answer: (A)

The key idea here is the pattern that defines the identity matrix and its role in matrix operations. An identity matrix is a square matrix with ones on the main diagonal and zeros everywhere else. It acts as the multiplicative identity, meaning multiplying any conformable matrix by it (on either side) leaves that matrix unchanged. For example, a 3x3 identity matrix looks like [ [1, 0, 0], [0, 1, 0], [0, 0, 1] ]. This pattern is what makes it the identity. If the diagonal entries aren’t all 1, or any off-diagonal entry is nonzero, the matrix won’t behave as the identity. A matrix with all ones isn’t the identity because it would alter other matrices under multiplication. A diagonal matrix with arbitrary diagonal values isn’t the identity unless those values are all 1. And a matrix with zeros on the diagonal and ones off the diagonal also changes other matrices when multiplied.

The key idea here is the pattern that defines the identity matrix and its role in matrix operations. An identity matrix is a square matrix with ones on the main diagonal and zeros everywhere else. It acts as the multiplicative identity, meaning multiplying any conformable matrix by it (on either side) leaves that matrix unchanged. For example, a 3x3 identity matrix looks like [ [1, 0, 0], [0, 1, 0], [0, 0, 1] ].

This pattern is what makes it the identity. If the diagonal entries aren’t all 1, or any off-diagonal entry is nonzero, the matrix won’t behave as the identity. A matrix with all ones isn’t the identity because it would alter other matrices under multiplication. A diagonal matrix with arbitrary diagonal values isn’t the identity unless those values are all 1. And a matrix with zeros on the diagonal and ones off the diagonal also changes other matrices when multiplied.