(6) Let A be a 200 × 200 matrix with determinant 3. what is the determinant of A
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Question
(6) Let A be a 200 × 200 matrix with determinant 3. what is the determinant of AT? (7) Let A be a 400 × 400 matrix with determinant 3, LetB be the matrix formed by swapping the second and third rows of A. What is det (A)? (8) Let A be a 20 × 20 matrix with determinant 3·Let B be the matrix formed by multiplying the fourth row of A by 2. What is det(B)? (9)Let A be a 100 x 100 matrix with determinant 3·Let B be the matrix formed by dividing the fourth row of A by 2. What is det(B)? (10) Let A be a 23 × 23 matrix with determinant 3·Let B be the matrix formed by adding two copies of the third row to the first. What is det(B)? (11) Let A be a 23 × 23 matrix with determinant 3·Let B be the matrix formed by subtracting two copies of the third row from the first. What is det(B)?Explanation / Answer
6) determinant will not change on tranposing the squre matrix so it will remain same
det(A)=det(AT)det(A)=det(AT) can be stated in a little different way.
Every square matrix can be represented as the product of an orthogonal matrix (representing an isometry) and an upper triangular matrix (QR decomposition)- where the determinant of an upper (or lower) triangular matrix is just the product of the elements along the diagonal (that stays in its place under transposition), so, by the Binet formula, A=QRA=QR gives:
det(A^T)=det(R*T.Q^T)
=det(R)det(Q^T)
=det(R)det(Q1),
det(A^T)=detR/detQ
=det(Q)det(R)
=det(QR)=det(A)
b) d according to theorem of squre matrix
that is If we swap two rows (columns) in A, the determinant will change its sign.
determinant of a will be remain same where as diterminant of b will be in opposite sign of determinant of a
8)determinant of b will be
det(b)=2.det(a)
9)determinant of b will be
det(b)=1/2.det(a)
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