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**Unformatted text preview: **The determinant of a square matrix is a real number calculated from the entries of the matrix that has the property of determining whether the matrix is invertible Sec 3.4 Saturday, February 06, 2010 11:18 PM Section3dot4 Page 1 The determinant of a square matrix is a real number calculated from the entries of the matrix that has the property of determining whether the matrix is invertible Example (i) A = [ 6 ] A is invertible if by row operations we can turn A into the identity, here I 1 = [ 1 ] Easy: R 1 ' = 1/6 R 1 gives A [ 1 ] , so [ 6 ] is invertible 3.4.1 Saturday, February 06, 2010 11:18 PM Section3dot4 Page 2 The determinant of a square matrix is a real number calculated from the entries of the matrix that has the property of determining whether the matrix is invertible Example (i) A = [ 6 ] A is invertible if by row operations we can turn A into the identity, here I 1 = [ 1 ] Easy: R 1 ' = 1/6 R 1 gives A [ 1 ] , so [ 6 ] is invertible Example (ii) A = [ 0 ] In this case we cannot turn A into the identity by row operations (why) CONCLUSION: A = [ a ] is invertible if and only if a ≠ 0 3.4.2 Saturday, February 06, 2010 11:18 PM Section3dot4 Page 3 The determinant of a square matrix is a real number calculated from the entries of the matrix that has the property of determining whether the matrix is invertible Example (i) A = [ 6 ] A is invertible if by row operations we can turn A into the identity, here I 1 = [ 1 ] Easy: R 1 ' = 1/6 R 1 gives A [ 1 ] , so [ 6 ] is invertible Example (ii) A = [ 0 ] In this case we cannot turn A into the identity by row operations (why) CONCLUSION: A = [ a ] is invertible if and only if a ≠ 0 The determinant of A = [ a ] is det( A ) = a Let A be a 1 x 1 matrix. If det( A ) ≠ 0, then A is invertible REMARK: The proof is essentially Example 1 3.4.3 (*) Saturday, February 06, 2010 11:18 PM Section3dot4 Page 4 a b The determinant of A = is det( A ) = ad - bc c d If A is a 2 x 2 matrix and if det( A ) ≠ 0, then A is invertible 3.4.4 Monday, February 15, 2010 8:24 AM Section3dot4 Page 5 a b The determinant of A = is det( A ) = ad - bc c d If A is a 2 x 2 matrix and if det( A ) ≠ 0, then A is invertible ( This follows from our formula in Section 3.2 for the inverse of a 2 x 2 matrix, but we will give an alternate proof here ) (see page 3.2.19) We are told that det( A ) ≠ 0, so this means that ad - bc ≠ 0 3.4.5 Monday, February 15, 2010 8:24 AM Section3dot4 Page 6 a b The determinant of A = is det( A ) = ad - bc c d If A is a 2 x 2 matrix and if det( A ) ≠ 0, then A is invertible ( This follows from our formula in Section 3.2 for the inverse of a 2 x 2 matrix, but we will give an alternate proof here ) (see page 3.2.19) We are told that det( A ) ≠ 0, so this means that ad - bc ≠ 0 NOTE: We may assume that a ≠ 0 REASON: If a = 0, then c ≠ 0, since otherwise we would have ad - bc = 0 which contradicts our assumption. So, if a = 0 0 b switch row 1 and row 2 a ' b ' with a ' ≠ 0 c d and relable the entries 0 d ' 3.4.6 Monday, February 15, 2010 8:24 AM Section3dot4 Page 7 a...

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