Problem 55 - Algebra and Trigonometry

Question

What is the multiplicative identity matrix?

Step-by-Step Solution

Answer

The multiplicative identity matrix is a special square matrix that leaves other matrices unchanged when they are multiplied by it. It's denoted by \(I\) and has 1s on the main diagonal and 0s elsewhere.

Step 1: Definition

The multiplicative identity matrix is a special type of square matrix. It is an array of numbers arranged in a square pattern, that when used in multiplication leaves the other matrix unchanged.

Step 2: Identifying its Structure

The identity matrix, often denoted as \(I\), has 1s on the main diagonal and 0s in all other places (elements outside the main diagonal). This is its unique property and definition.

Step 3: Presenting an Example

For instance, the 3x3 identity matrix is: \[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\] Any matrix multiplied by its corresponding identity matrix equals the matrix itself.