Problem 55 - College Algebra

Question

What is the multiplicative identity matrix?

Step-by-Step Solution

Answer

The multiplicative identity matrix is a square matrix, which has ones on its principal diagonal (from top-left to bottom-right) and zeros everywhere else. When it multiplies any matrix, the resulting matrix is the original matrix.

Step 1: Understand Identity Matrix

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. It's denoted by \(I_n\) or simply \(I\), where \(n\) is the order of the square matrix.

Step 2: Structure of Identity Matrix

The identity matrix takes a simple form: it's a square matrix (which means it has the same number of rows as columns), with 1s on the diagonal and 0s everywhere else. For example, an identity matrix of order 3, \(I_3\) is represented as: \[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{bmatrix}\]

Step 3: Properties of Identity Matrix

When identity matrix is multiplied with any matrix, the result is the original matrix itself. For example, if \(A\) is any matrix and \(I\) is an identity matrix, then \(A \times I = A\) and \(I \times A = A\). This is similar to multiplying numbers by 1; the product is the original number. The identity matrix \(I\) functions as a multiplicative identity for matrices.