Practice Makes Perfect Algebra - Carolyn Wheater [36]
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Determinants
The determinant of a matrix is a single number associated with the matrix. Although that is a rather uninformative definition, it is difficult to give a better one. In spite of this difficulty, determinants are important in our study of matrix arithmetic.
Only square matrices have determinants, and in the simplest case, the determinant of a 1 × 1 matrix is the single element of the matrix. The determinant of a matrix A is indicated as |A|. The bars that indicate the determinant may remind you of the symbols for absolute value, but the significance is quite different. We can use the bars around the entire array rather than the name. The determinant of the matrix is denoted by .
Finding the determinant of a 2 × 2 matrix
In a square matrix, the diagonal path from upper left to lower right is called the major diagonal. The diagonal from upper right to lower left is the minor diagonal. In the matrix , the major diagonal contains the elements 4 and 1, while the minor diagonal contains 8 and 3.
The determinant of a 2 × 2 matrix is equal to the product of the elements on the major diagonal minus the product of the elements on the minor diagonal. The determinant . If we write this in symbolic terms, we can say The determinant of the square matrix can be found quickly by applying the rule:
Finding the determinant of a 3 × 3 matrix
To find the determinant of a 3 × 3(or larger) matrix, we can follow a plan called expanding along a row or column. Because this process can be cumbersome, most people turn to technology for large determinants. We will investigate the expansion, however, because it allows us to develop a formula for the determinant of a 3 × 3 matrix, which can sometimes be useful.
Begin with a general expression for the determinant of a 3 × 3 matrix: . We choose a row or column along which to expand. There are some sign changes in the process depending on which row or column we choose. For this example, we will expand along the top row. Imagine three copies of this determinant, but in each copy, circle one element of the top row and then cross out the rest of the top row and the rest of the column containing the circled element. The three versions should look like this.
In each version, you see four elements untouched and forming a 2 × 2 matrix. Then we expand the 3 × 3 determinant as
Notice that in each step we multiply the circled element times the determinant of the remaining 2 × 2 matrix. Note, too, that we alternate adding and subtracting. In the last section, we developed a simple rule for the determinant of a 2 × 2 matrix, so we can apply it here.
Larger determinants are extremely tedious to compute. To find a 4 × 4 determinant, for example, we would first expand to sums and differences of products of elements and 3 × 3 determinants. Those 3 × 3 determinants would then be expanded, and the result involves much arithmetic. Technology simplifies the process tremendously.
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EXERCISE 13.5
Find each determinant, if possible.
Find the missing element.
Find each determinant.
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Inverses
You may have noticed in our discussion of matrix arithmetic that no mention was made of division. While there is no operation of matrix division, the arithmetic of matrices does include the concept of a multiplicative inverse. The idea of the inverse is familiar from standard arithmetic, even if that term is not used as commonly.
In arithmetic, you learned that every non-0 number has a reciprocal and that the product of the number and its reciprocal is 1. The reciprocal of , for example, is , and the product . What we commonly refer to as the reciprocal is formally called the multiplicative inverse. Two numbers are multiplicative inverses if their product is 1, the identity element for multiplication.
To transfer the concept of multiplicative inverse to matrix arithmetic, we need to establish