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Theorem 2.2.2 An n × n matrix A is singular if and only if det(A) = 0 Proof The matrix A can be reduced to row echelon form with a finite number of row operations. Thus, U = E_{k}E_{k-1} ? E_{1}A where U is in row echelon form and the E_{i}’s are all elementary matrices. It follows that det(U) = det(E_{k}E_{k-1} ? E_{1}A) = det(E_{k}) det(E_{k-1}) ? det(E_{1}) det(A) Since the determinants of the E_{i}’s are all nonzero, it follows that det(A) = 0 if and only if det(U) = 0. If A is singular, then U has a row consisting entirely of zeros, and hence det(U) = 0. If A is nonsingular, then U is triangular with 1’s along the diagonal and hence det(U) = 1. From the proof of Theorem 2.2.2, we can obtain a method for computing det(A). We reduce A to row echelon form. U = E_{k}E_{k-1} ? E_{1}A If the last row of U consists entirely of zeros, A is singular and det(A) = 0. Otherwise, A is nonsingular and det(A) = [det(E_{k}) det(E_{k-1}) ? det(E_{1})]^{-1} Actually, if A is nonsingular, it is simpler to reduce A to triangular form. This can be done using only row operations I and III. Thus, T = E_{m}E_{m-1} ? E_{1}A and hence, det(A) = ± det(T) = ± t_{11}t_{22} ? t_{nn} where the t_{ii}’s are the diagonal entries of T. The sign will be positive if row operation I has been used an even number of times and negative otherwise.

          Theorem 2.2.2 An n × n matrix A is singular if and only if

det(A) = 0

Proof The matrix A can be reduced to row echelon form with a finite number of row operations. Thus,

U = E_{k}E_{k-1} ? E_{1}A

where U is in row echelon form and the E_{i}’s are all elementary matrices. It follows that

det(U) = det(E_{k}E_{k-1} ? E_{1}A)
= det(E_{k}) det(E_{k-1}) ? det(E_{1}) det(A)

Since the determinants of the E_{i}’s are all nonzero, it follows that det(A) = 0 if and only if det(U) = 0. If A is singular, then U has a row consisting entirely of zeros, and hence det(U) = 0. If A is nonsingular, then U is triangular with 1’s along the diagonal and hence det(U) = 1.

From the proof of Theorem 2.2.2, we can obtain a method for computing det(A). We reduce A to row echelon form.

U = E_{k}E_{k-1} ? E_{1}A

If the last row of U consists entirely of zeros, A is singular and det(A) = 0. Otherwise, A is nonsingular and

det(A) = [det(E_{k}) det(E_{k-1}) ? det(E_{1})]^{-1}

Actually, if A is nonsingular, it is simpler to reduce A to triangular form. This can be done using only row operations I and III. Thus,

T = E_{m}E_{m-1} ? E_{1}A

and hence,

det(A) = ± det(T) = ± t_{11}t_{22} ? t_{nn}

where the t_{ii}’s are the diagonal entries of T. The sign will be positive if row operation I has been used an even number of times and negative otherwise.

Theorem 2.2.2 An n × n matrix A is singular if and only if

det(A) = 0

Proof The matrix A can be reduced to row echelon form with a finite number of row operations. Thus,

U = EkEk-1 ? E1A

where U is in row echelon form and the Ei’s are all elementary matrices. It follows that

det(U) = det(EkEk-1 ? E1A)
= det(Ek) det(Ek-1) ? det(E1) det(A)

Since the determinants of the Ei’s are all nonzero, it follows that det(A) = 0 if and only if det(U) = 0. If A is singular, then U has a row consisting entirely of zeros, and hence det(U) = 0. If A is nonsingular, then U is triangular with 1’s along the diagonal and hence det(U) = 1.

From the proof of Theorem 2.2.2, we can obtain a method for computing det(A). We reduce A to row echelon form.

U = EkEk-1 ? E1A

If the last row of U consists entirely of zeros, A is singular and det(A) = 0. Otherwise, A is nonsingular and

det(A) = [det(Ek) det(Ek-1) ? det(E1)]^-1

Actually, if A is nonsingular, it is simpler to reduce A to triangular form. This can be done using only row operations I and III. Thus,

T = EmEm-1 ? E1A

and hence,

det(A) = ± det(T) = ± t11t22 ? tnn

where the tii’s are the diagonal entries of T. The sign will be positive if row operation I has been used an even number of times and negative otherwise.

Added by Leslie A.

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