Linear Algebra

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-1A = 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 = BTAT

• (AB)-1 = B-1A-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.