Determinants Flashcards
Determinant of a 2 x 2 matrix
det A = ad-bc
Define the determinant of a 3 x 3 matrix in terms of the columns
A =
The matrix A fails to be invertible if the image of A isn’t all of R^3, meaning that the three column vectors u,v, and w are contained in some plan V. In this case, the cross product v x w, being perpendicular to V, is perpendicular to vector u so that
u(dot)(v x w) = 0
However, if A is invertible, then v x w doesnt equal zero (ie not perpendicular) so u(dot)(v x w) /=0
Therefore, a 3 x 3 matrix A is invertible if(and only if) detA=u(dot)(v x w) /= 0
What is sarrus’ rule?
A way to find the determinant of a 3 x 3 matrix. Multiply the diagonals of the matrix A by applying the first two columns of A on the right side of the matrix A
What is the determinant of a triangular matrix?
Any upper or lower triangular matrix’s determinant will be equivalent to the product of their diagonal entries
What is a transpose?
The transpose of a given matrix A (denoted A^T) is the n x n matrix whose ijth entry is the jith entry of A: The roles of rows and columns are reversed
What is the det(A^T)?
det(A^T) = det(A)
Gauss Jordan Elimination and Determinants
What happens to the det B when:
a. B is obtained from A by dividing a row of A by a scalar k
b. If B is obtained from A by a row swap
c. If B is obtained from A by adding a multiple of a row of A to another row
a. detB=(1/k)detA
b. detB= -detA
c. detB = detA
What makes a square matrix A invertible?
A square matrix A is invertible if and only if detA does not (/=) 0
If you reduce an invertible n x n matrix A to I, what will be the detA?
detA=(-1)^(number of row swaps)*(k1k2…kr) where k1,k2,…,kr are scalars that you divide various rows by
If it is too tedious to reduce A to I, what must you do in order to determine the detA?
Reduce A to some matrix B whose determinant is easy to compute (a triangular matrix possibly) then the determinant will be:
detA = (-1)^(number of row swaps)*(k1k2…kr)detB
What is det(AB)?
det(AB) = det(A)det(B)
What is det(A^m)?
det(A^m)=(detA)^m
If matrix A is similar to matrix B, what is the relationship between the detA and detB?
By definition of similar matrices, there exists a matrix S where AS = SB. Now we have:
(detA)(detS) = (detS)(detB)
divide both sides by nonzero scalar detS
detA = detB
If A is an invertible n x n matrix, what is the relationship between detA and det(A^-1)?
By definition of an inverse, the equation I = AA^-1 holds. by taking the determinant of both sides we find
1 = det(I) = det(AA^-1) = det(A)det(A^-1)
so that det(A^-1) = 1/det(A) = (detA)^-1
