Problem 49 - College Algebra

Question

Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.

Step-by-Step Solution

Answer

A matrix that does not have the same number of rows and columns, or a non-square matrix, cannot have a multiplicative inverse because the definition of the multiplicative inverse requires the multiplication of the matrix and its inverse to result in the Identity matrix, which is a square matrix. Thus, non-square matrices can never generate a square Identity matrix via multiplication, implying that they can't possess a multiplicative inverse.

Step 1: Definition of a Square Matrix

A square matrix is a type of matrix that has the same number of rows as columns. In a square matrix, if you start from any cell, you can reach any other cell by going through the appropriate number of rows and columns. For example, a 3x3 matrix is a square matrix because there are 3 rows and 3 columns.

Step 2: Concept of a Multiplicative Inverse

In the language of matrices, the multiplicative inverse (or simply, the 'inverse') of a matrix A is often denoted as \( A^{-1} \). It has the property where when it is multiplied with the original matrix, we get the Identity matrix (I), i.e. \( A * A^{-1} = A^{-1} * A = I \), where I is the Identity matrix that has 1s on its leading diagonal (top left to bottom right) and 0s everywhere else.

Step 3: Linking Square Matrices and the Inverse

Non-square matrices do not have multiplicative inverses. This is because of the aforementioned definition – the multiplication of a matrix and its inverse results in the Identity matrix, which is inherently a square matrix. When the original matrix is not square, its multiplication (regardless with which other matrix) will not yield a square matrix, hence eliminating the possibility of the result being an Identity matrix. Therefore, only square matrices have a chance of incurring an Identity matrix as their product with another matrix and hence are the only matrices that can possibly have a multiplicative inverse.