Prentice Hall's Online Lesson Quiz Enter Web Code: aga–0405
Every square matrix has a number associated with it, called its determinant. A determinant is a square array of numbers or variables enclosed between two parallel vertical bars. The numbers or variables written within a determinant are called elements.
The determinant of a 2 × 2 matrix has two rows and two columns and is called a second-order determinant . A second-order determinant is evaluated as follows.
The determinant of the 2 × 2 matrix is denoted by the symbol
and is defined as follows.
Notice that the value of the determinant is found by calculating the difference of the products of the two diagonals.
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Recall from working with real numbers that the inverse and identity of a number are related. In real numbers, 1 is the identity for multiplication because a · 1 = 1 · a = a. Similarly, the inverse of a matrix is related to the identity matrix.
Another property of real numbers is that every real number, except 0, has a multiplicative inverse. That is,
is the multiplicative inverse of a because
. We will explore both Identity and Inverse matrices in this section.
The identity matrix is a square matrix that, when multiplied by another matrix, equals that same matrix.
With 2 × 2 matrices,
is the identity matrix. The identity matrix is symbolized by I.
Since one of the properties of the identity matrix is that it is commutative, only a square matrix can have an identity. In an identity matrix, the principal diagonal goes from upper left to lower right and consists only of ones.
The inverse of matrix A is written
(read "A inverse"). If matrix A has an inverse named
. The example that follows shows how the inverse of a 2 × 2 matrix can be found.
example:
Given B, find
Let. By definition,
.
Multiply.
When two matrices are equal, their corresponding elements are equal. So the following equations can be generated from the two equal matrices.
Use equations (1) and (3) to find values for w and y.
Therefore by substitution, w = 4.
Use equations (2) and (4) to find values for x and z.
Therefore by substitution, x = -3.
The inverse matrix is shown below.
Check:
A Formula for the Inverse of a 2 × 2 Matrix
Inverse of a 2 × 2 MatrixAny matrix
will have an inverse
if and only if
Then
Cryptology or code theory offers an interesting application of matrix inverses, at least to governments. One of the most famous accounts of code theory comes from World War II. A group of American Indians called the Navaho Code Talkers, directed the entire military operation at Iwo Jima through orders communicated by the Code Talkers. The Japanese were never able to break the code.
In the 1930s, an American mathematician named Lester Hill made important advances in cryptology by using matrices to encode messages. Today, programmers use code theory to protect secret data stored in computers. Here is a simplified version of how it works.
Suppose our message is MATH ROCKS.
Step 1:
For simplicity, let each letter of the alphabet correspond to a number based on its position (A = 1, B = 2, C = 3, and so on). Ignore punctuation and let 27 correspond to a blank space. Break the message into groups of two or three letters in order to use an encoding matrix. Write the numbers in a matrix. Here we chose 2 rows and 1 column.
Step 2:
Now choose any matrix that has an inverse to use for encoding. Multiply the matrix by a coding matrix. We will use the following:
The entries of the product matrices can now be transmitted as the message shown below. When numbers are greater than 27, simply repeat the alphabet. When numbers are negative reverse the order. For example 0 = space, -1 = Z, -2 = Y, -3 = X, etc.
ANHARRCRSCStep 3:
When the person receives the message, he needs to decipher it by undoing the multiplication. Mathematically, this means to find the inverse of the coding matrix.
First, write the letters of the message as 2 × 1 matrices. Then multiply each matrix by the inverse matrix. Finally assign letters to the elements in the product.
