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# 2: Matrices

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• 2.1: Matrix Arithmetic
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.
• 2.2: Multiplication of Matrices
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.
• 2.3: The ijth Entry of a Product
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.
• 2.4: Properties of Matrix Multiplication
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.
• 2.5: The Transpose
Another important operation on matrices is that of taking the transpose.
• 2.6: The Identity and Inverses
There is a special matrix, denoted I , which is called to as the identity matrix
• 2.7: Finding the Inverse of a Matrix
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$$^−1$$.
• 2.8: Elementary Matrices
We now turn our attention to a special type of matrix called an elementary matrix.
• 2.9: More on Matrix Inverses
In this section, we will prove three theorems which will clarify the concept of matrix inverses.
• 2.10: LU
An $$LU$$ factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix $$L$$ which has the main diagonal consisting entirely of ones, and an upper triangular matrix $$U$$ in the indicated order.
• 2.E: Exercises

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