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12.1: Matrix Arithmetic
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You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications.
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12.2: Multiplication of Matrices
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The next important matrix operation we will explore is multiplication of matrices. The operation of matrix multiplication is one of the most important and useful of the matrix operations.
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12.3: The ijth Entry of a Product
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In previous sections, we used the entries of a matrix to describe the action of matrix addition and scalar multiplication. We can also study matrix multiplication using the entries of matrices.
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12.4: Properties of Matrix Multiplication
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As pointed out above, it is sometimes possible to multiply matrices in one order but not in the other order. However, even if both AB and BA are defined, they may not be equal.
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12.5: The Transpose
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Another important operation on matrices is that of taking the transpose.
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12.6: The Identity and Inverses
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There is a special matrix, denoted I , which is called to as the identity matrix
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12.7: Finding the Inverse of a Matrix
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In Example 2.6.1, we were given A^\(−1\) and asked to verify that this matrix was in fact the inverse of A. In this section, we explore how to find A\(^−1 \).
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12.8: Basic Techniques of Determinants
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Let A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
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12.9: Properties of Determinants
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There are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of row operations on the determinant of a matrix. In future sections, we will see that using the following properties can greatly assist in finding determinants. This section will use the theorems as motivation to provide various examples of the usefulness of the properties.
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12.10: Finding Determinants using Row Operations
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In this section, we look at two examples where row operations are used to find the determinant of a large matrix.
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12.E: Exercises
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