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MATH 1ZC3 (7)


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Chris Mc Lean

HOMOGENEOUS SYSTEMS  if all equations are homogeneous, matrix must be consistent o All have the trivial solution (x1 = 0, x2 = 0, …) o They may have additional solutions as well  in this case, they have infinitely many solutions o If system has more unknowns than equations, it has infinitely many solutions  thus has a non-trivial solution  If a homogeneous linear system has n unknowns, and r non-zero rows, it has n-r free variables MATRIX ADDITION  A+B=B+A  A+(B+C)=(A+B)+C MATRIX MULTIPLICATION  A(BC) = (AB)C  A(B+C) = AB+AC  (B-C)A = BA-CA  a(B+C) = aB+ aC  (a+b)C = ac+bc  a(bC) = (ab)C  a(BC) =(aB)C = B(aC) TRACES  The trace of A is undefined if A is not square  The trace is the sum of entries on the principal diagonal  tr A = tr(A)  tr(A+B) = tr(A) + tr(B)  tr(cA) = c tr(A)  tr(AB) = tr(BA) TRANSPOSES  (A ) = A T T T  (A+B) =A + B  (kA) = kA T T T T  (AB) = B A  (A )-1 = (A ) T T T  A A and AA are symmetric, and if A is invertible, they are also invertible INVERSES -1 -1 -1  (AB) = A B  (A ) = A = (A )1 n -1 -1  (A ) = A  (kA) = k A -1  A diagonal matrix is invertible only if all of its diagonal entries are non-zero  A matrix is invertible as long as its determinant isn’t zero  If A and B are square, same dimensions, and AB is invertible, A and B are each invertible  If A is a singular, square matrix, Ax=0 has infinitely many solutions  If A is invertible, and the first row is added to the second, A is still invertible DETERMINANTS  A is invertible only if det(A) isn’t 0 -1  det(A ) = 1/det(A)  det(cA) = c^n(det(A))  det(A^n) = (det(A))^n T  det(A) = det(A )  det(AB) = det(A)det(B) T T  det(A+B ) = det(A + B)  det(A +B ) = det(A +B) TRIANGULAR MATRICES  Transpose o Transpose LOWER = UPPER o Transpose UPPER = LOWER  multiply o product of LOWERS = LOWER o product of UPPERS = UPPER  Inverse o Inverse of LOWER = LOWER o Inverse of UPPER = UPPER SYMMETRY  If A and B are nxn and symmetric o A+B is symmetric o A-B is symmetric o kA is symmetric  AB is symmetric ONLY if the matrices COMMUTE -1  If A is symmetric, invertible, A is symmetric T T T T  if A A and AA are symmetric, and A is invertible, A A and AA are also invertible  the product of SKEW SYMMETRIC matrices that commute is symmetric  If A and B are symmetric, AB + BA is sy
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