MATRIX ALGEBRA. MGT 4850 Spring 2008 University of Lethbridge. Laws of Arithmetic. Let A,B,C be matrices of the same size m × n, 0 the m × n zero matrix, and c and d scalars. (1) (Closure Law) A + B is an m × n matrix. (2) (Associative Law) ( A + B) + C = A + (B + C)
University of Lethbridge
2x − 3y + 4z = 5
as a “product” of the coefficient matrix
and the column matrix of unknowns
⎡ x ⎤
⎣ z ⎦
(1) (Closure Law) The product AB is a matrix.
(2) (Associative Law) (AB)C = A(BC)
(3) (Identity Law) AI = A and IB = B
(4) (Associative Law for Scalars) c(AB) = (cA)B = A(cB)
(5) (Distributive Law) (A + B)C = AC + BC
(6) (Distributive Law) A(B + C) = AB + AC
Any nonsquare matrix is noninvertible. Square matrices are classified as either “singular,” i.e., noninvertible, or nonsingular,” i.e., invertible. Since we will mostly be concerned with two-sided inverses, the unqualified term “inverse” will be understood to mean a “two-sided inverse.” Notice that
this definition is actually symmetric in A and B. In other words, if B is an inverse for A, then A is an inverse for B.
(1) (Uniqueness) If A is invertible, then it has only one inverse, by A−1.
(2) (Double Inverse) If A is invertible, then (A−1)−1 = A.
(3) (2/3 Rule) If any two of the three matrices A, B, and AB are invertible, then so is the third, and moreover, (AB)−1 = B−1A−1.
(4) If A is invertible, then (cA)−1 = (1/c)A−1.
(5) (Inverse/Transpose) If A is invertible, then (AT )−1 = (A−1)T .
(6) (Cancellation) Suppose A is invertible. If AB = AC or BA = CA, then B = C.
skip from p.103 to p.113