Problem 47 - College Algebra

Question

What is the multiplicative identity matrix?

Step-by-Step Solution

Answer

The multiplicative identity matrix is a square matrix with ones on its diagonal and zeros everywhere else. When it is multiplied by any other matrix, the result is still the original matrix.

Step 1: Definition of an Identity Matrix

The multiplicative identity matrix, often simply called the identity matrix, is a square matrix (same number of rows as columns), denoted as I, with ones on the diagonal and zeros everywhere else. For a 2x2 matrix, it is: \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

Step 2: Properties of an Identity Matrix

One of the important properties of an identity matrix is that when it is multiplied by any other matrix, the original matrix is unchanged. It's symbolized as: If A is any m×n matrix, then IA = A and AI = A.

Step 3: Example of Multiplying with an Identity Matrix

For instance, if we have a matrix A such as: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] And multiply it with the 2x2 identity matrix I, we will still get the same matrix A. Like so: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} * \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]