Let 0n be the zero vector of length is nonsingular if and only if 0n is n. A the only solution of Ax=0n. is nonsingular if and only if det(A)0. A ! such that if and only if is nonsingular. Lemma: A-1 A-1A=AA-1=I A Suppose such that but Then Proof: C CA-1, CA=AC=I. C=IC=(A-1A)C=A-1(AC)=A-1I=A-1. Diagonal matrices: . Triangular matrices: upper , strictly upper . lower , strictly lower . 3b. Gaussian elimination
Let 0n be the zero vector of length n. A is nonsingular if and only if 0n is the only solution of Ax=0n. A is nonsingular if and only if det(A)0. Lemma: ! A-1 such that A-1A=AA-1=I if and only if A is nonsingular. Proof: Suppose C such that CA-1, but CA=AC=I. Then C=IC=(A-1A)C=A-1(AC)=A-1I=A-1. Diagonal matrices: . Triangular matrices: upper , strictly upper . lower , strictly lower . 3b. Gaussian elimination
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