Problem 57 - 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 must be square in order to have an inverse because it needs to have a non-zero determinant, which can only be computed for square matrices, and it needs to be able to multiply with its inverse to form the identity matrix, which requires the number of rows of the first to equal the number of columns in the second.

Step 1: Understand the inverse of a matrix

The inverse of a matrix A is another matrix, denoted as A^-1, such that when A is multiplied with A^-1, the result is the identity matrix. The identity matrix is a special square matrix with 1's on the diagonal and 0's elsewhere.

Step 2: Express the requirement for an inverse matrix

For a matrix to have an inverse, it must satisfy two conditions: 1) It must be square (i.e., have the same number of rows and columns). 2) Its determinant must be non-zero. A non-square matrix will not have a determinant, and thus an inverse cannot be defined for it.

Step 3: Connect the square requirement to multiplication

Matrix multiplication is only possible if the number of columns of the first matrix is equal to the number of rows of the second one. Hence, when attempting to multiply a matrix by its inverse to get the identity matrix (which is always square), both the original matrix and its inverse must be square matrices. Otherwise, the multiplication would not be possible.