Read: Boas Ch. 3.

**9.1 Properties of and operations with matrices
**

M × N matrix with elements

Definitions:

• Matrices are equal if their elements are equal,
.

•

• for k const.

• : Note for multiplication of rectangular
matrices, need

(M × N) · (N × P).

• Matrices need not "commute". AB not nec. equal to BA. [A, B] ≡ AB -BA

is called "commutator of A and B. If [A,B] = 0, A and B commute.

• For square mats. N×N, , where

sum is taken over all permutiations π of the elements {1, ...N }. Each term

in the sum is a product of N elements, each taken from a different row of A

and from a different column of A, and sgn π. Examples:

• detAB = detA · detB but det(A + B) ≠ detA + detB. For
practice with

determinants, see Boas.

• Identity matrix I:
.

• Inverse of a matrix. A · A^{-1} = A^{-1}A = I.

• Transpose of a matrix

• Formula for finding inverse:

where C is "cofactor matrix". An element
is the
determinant of the N -

1 × N - 1 matrix you get when you cross out the row and column (i,j), and

multiply by (-1)^{i}(-1)^{j} . See Boas.

• Adjoint of a matrix. is adjoint of
A, has elements , i.e. it's

conjugate transpose. Don't worry if you don't know or have forgotten what

conjugate means.

• (AB)^{T} = B^{T}A^{T
}

• (AB)^{-1} = B^{-1}A^{-1
}

• "row vector" is 1 × N matrix: [a b c ... n]

• "column vector" is M × 1 matrix:

• Matrix is "diagonal" if
.

• "Trace" of matrix is sum of diagonal elements:. Trace of

produce is invariant under cyclic permutations (check!):

**9.2 Solving linear equations**

Ex.

x - y + z = 4

2x + y - z = -1

3x + 2y + 2z = 5

may be written

Symbolically, . What we
want is . So
we find the determinant

detA = 12, and the cofactor matrix

then take the transpose and construct .

so

So x = 1, y = -1, z = 2. Alternate method is to use
Cramer's rule (see Boas p.

93):

and "similarly" for y and z. Here, the 12 is detA, and the
determinant shown is

that of the matrix of coefficients A with the x coefficients (in this case) replaced

by

Q: what happens if = 0? Eqs. are homogeneous ) => detA = 0.

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