9.4 Multiplicative Inverses of Matrices and Matrix Equations

How to Encode a Word or Message Using a Matrix

1. Express the word or message numerically. 2. List the numbers in step 1 by columns and form a square matrix. if you do not have enough numbers to form a square matrix, put zeros in any remaining spaces in the last column. 3. Select any square invertible matrix, called the coding matrix, the same size as the matrix in step 2. Multiply the coding matrix by the square matrix that expresses the message numerically. The resulting matrix is the coded matrix. 4. Use the numbers, by columns, from the coded matrix in step 3 to write the encoded message.

How to Use a Graphing Utility to Solve a Linear System with the Inverse of a Matrix System

Enter the elements in A (the coefficient matrix), and B (the column matrix). The find the product of A inverse and B.

How do you find the Multiplicative Inverse of a Square Matrix?

Find both AB and BA. If B is the multiplicative inverse of A, both products (AB and BA) will be the multiplicative identity matrix, In

Do all matrices have an inverse?

No. A nonsquare matrix, one with a different number of rows than columns, cannot have a multiplicative inverse. While only square matrices can have multiplicative inverses, it is not the case that every square matrix possesses a multiplicative inverse. (see image)

How to Decode a Word or Message

The inverse of a coding matrix can be used to decode a word or message that was encoded. 1. Find the multiplicative inverse of the coding matrix. 2. Multiply the multiplicative inverse of the coding matrix and the coded matrix. 3. Express the numbers, by columns, from the matrix in step 2 as letters.

How to Solve Linear Systems of Equations Using Multiplicative Inverses of Matrices

The matrix equation is abbreviated as AX = B, where A is the coefficient matrix of the system, and X and B are matrices containing one column, called column matrices. The matrix B is called the constant matrix.

What is the Multiplicative Identity Matrix?

The n x n square matrix with 1s down the main diagonal from upper left to lower right and 0s elsewhere is called the multiplicative identity matrix of order n.

Quick Method for Finding the Multiplicative Inverse of a 2x2 matrix

When using this formula to find the multiplicative inverse of a 2x2 matrix, start by computing ad - bc. If the computed value is 0, there is no need to continue. The given matrix is singular (it has no multiplicative inverse). Reverse a and d and negate b and c.