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 If the system has fewer eqns than variables, then it must have an infinite number of solns if a matrix possesses an inverse, then the inverse is unique. Inverse of A is A^-1 Because the system have the same coeff matrix one could join the two non-coeff column matrices together If the system has fewer eqns than variables, then it must have an infinite number of solns the sum of two matrices of diff sizes is undefined, b/c the # of columns must be equal to the # of rows in the 2nd matrix Singular matrix results in a row of zeros, which makes it impossible to rewrite into [I|A^-1] what is a symmetric matrix square matrix that is equal to its transpose matrix (A^T)^T formula for inverse of A matrix is A^-1 =1/ad-bc[d -b] [-c a] if det is not equal to zero (AB)^-1 A^-1 *B^-1 if c is invertible, then cancellation rule applies 1. if AC=BC, then A=B 2. if CA=CB, then A=B if A is an invertible matrix, the sys of eqns Ax=b has a unique soln given by x=A^-1b two eqns are called parallel if the left side of one eqn is a multiple of the other and inconsistent polynomial in one variable of degree 2 is called p(x) = a0 + a1x + a^2*x^2 y=a0+a1x+a2x2 det(AB) det(A)*det(B) An nxn matrix is invertible<-> determinant is not equal to zero prove by induction statemts s1,s2,s3,...Sn -show S1 is true -show if Sn holds, then so does next one(Sn+1) -Conclusion:Sn is true for al n an nxn matrix is called elementary matrix if it can be obtained from the identity matrix by a single elementary row operation and it needs to be a square matrix every elementary matrix is invertible and the inverse of an elementary matrix is another elementary mateix a square matrix A is invertible<-> it can be written as the product of elementary matrices a square matrix A is invertible<-> it can be written as the product of elementary matrices