This is the second of two weeks devoted to solving systems of linear equations. It begins with two sections on matrices
after which we cover one more method for solving linear equations:
We will skip section 4-7.
An mxn matrix is a rectangular array of mn numbers, arranged in m rows of n numbers each. Matrices can be added, subtracted, and multiplied by numbers or by other matrices, as long as the matrix sizes are compatible.
"Compatible" means different things for different operations: to add or subtract two matrices, they have to be of the same size. To multiply two matrices, their "inner dimensions" have to agree. If A is an mxn matrix, and B is an nxp matrix, and you put the dimensions side by side: (mxn)(nxp), the "inner dimension" is n, and the "outer dimensions" are m and p. The inner dimension disappears, and the result is of size mxp (the outer dimensions).
We use the notation 0 for the zero matrix, which is a matrix full of zeros, and talk about the zero matrix, even though technically there are lots of different zero matrices: a zero matrix of size 2x3, a zero matrix of size 4x2, etc. The zero matrix corresponds to the number 0. For example, A0 = 0A = 0 for any A, A+0 = A, etc.
If the product AB exists, the product BA does not necessarily also exist. Even if it does, it does not have to be of the same size: if A is of size 2x3, and B is of size 3x2, then AB has size 2x2, and BA has size 3x3. Even if A and B are square matrices of the same size, so that both AB and BA are also of the same size, they don't have to be the same: matrix multiplication is not commutative.
Another property of matrix multiplication which is different from multiplying numbers is that it is possible to have two non-zero matrices A, B with AB=0.
Except for the properties mentioned in the last two paragraphs, matrix arithmetic satisfies all the usual properties of standard arithmetic. For example, A(B+C) = AB + AC, A(BC) = (AB)C, and so on.
The identity matrix of size nxn is the matrix I which has 1 on the principal diagonal, 0 everywhere else. It has the property that IA = A and AI = A, so I corresponds to the number 1. As with the zero matrix, there are really lots of different identity matrices, one for each (square) size.
If you can find a pair of matrices A, B with AB=I and BA=I, then B is called the inverse of A, written A-1. Every number except 0 has an inverse (the inverse of 2 is (1/2)), but not every matrix has an inverse. However, if an inverse exists, it is unique.
Inverse matrices can be used to solve matrix equations. If AX=B, then X=A-1B. Pay attention to the order: X=BA-1 is the solution to XA = B, which is not the same.
A more complicated example is this: Solve AX+X=B.
Solution: AX+X = AX+IX = (A+I)X = B, so B=(A+I)-1B.
Read the textbook and do the homework assignment HW 5.