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6.2 Properties of Determinants. Finding a determinant using row reductions. One can find the determinant of a matrix by performing row reductions. With the following properties: If B is obtained from A by dividing a row of A by a scalar k then det(B) =1/k(det(A)) OR kdet(B) =det(A)
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Finding a determinant using row reductions One can find the determinant of a matrix by performing row reductions. With the following properties: • If B is obtained from A by dividing a row of A by a scalar k then det(B) =1/k(det(A)) OR kdet(B) =det(A) 2) If B is obtained from A by a row swap then det(B) = -det(A) 3) If B is obtained from A by adding a multiple of one row to another row then det(B) = det(A)
Example 1 Find the determinant of the matrix by row reductions
Problem 11 Solution • Det(A) = 8 Det(B) = (8)(-9) = -72
To determine if a matrix is singular A square matrix is invertible if and only if Det(A)≠0 A square matrix is singular if and only if Det(A)=0
Example 3 • Determinant of a productdet(AB) = det(A)*det(B) Use this fact to show that there is no matrix such that
Determinant of the Transpose of a Matrix Why is this true?