Consider A, B and C, all nxn matrices. Show that: 1) det(A)=det(A^T) 2) if C was...

Consider A, B and C, all nxn matrices.
Show that:
1) det(A)=det(A^T)
2) if C was obtained from A by changing the i-th row (column) with the j-th row (column). Show that det(C)=-det(A)
3) det(AB)=det(A)det(B)
4) Let C be a matrix obtained from A by multiplying a row by c ∈ F. Show that det(B)=c · det(A)

Expert Solution

Colby Messinger answered 2 years ago

Consider only square matrices. True or false? Explain. a) det(AB) = (det A)(det B). b) Eigenvalues...

Consider only square matrices. True or false? Explain. a) det(AB) = (det A)(det B). b) Eigenvalues of all real matrices are real. c) The determinant of an upper triangular matrix is the sum of its main diagonal entries.

Given the nxn matrices A,B,C of real numbers, which satisfy the Condition: A+B+λΑΒ=0 Β+C+λBC=0 A+C+λCA=0 for...

Given the nxn matrices A,B,C of real numbers, which satisfy the Condition: A+B+λΑΒ=0 Β+C+λBC=0 A+C+λCA=0 for some λ≠0 ∈ R (α) Prove that I+λΑ,Ι+λΒ,Ι+λC are invertible and AB=BC=CA. (b) Prove that A=B=C

verify that the function det has the following properties 1. det(... ,v,..., v,...)=0 2. det(...,cv+dw,...)=cdet(...,v,...)+ddet(...,w,...) 3....

verify that the function det has the following properties 1. det(... ,v,..., v,...)=0 2. det(...,cv+dw,...)=c*det(...,v,...)+d*det(...,w,...) 3. use the above properties and normalization to prove that A= det(...,v+cw,...w,...)=det(...,v,...,w,...) B=det(...,v,...,w,...)= (-1)*det(...,w,...,v,...) and C= det [diag (λ1, ... , λn ) ] = Πiλi

for square matrices A and B show that [A,B]=0 then [A^2,B]=0

a+b+c=(abc)^(1/2) show that (a(b+c))^(1/2)+(b(c+a))^(1/2)+(c(a+b))^(1/2)>36

a) Show that if A is a real, non-singular nxn matrix, then A.(A^T) is positive definite....

a) Show that if A is a real, non-singular nxn matrix, then A.(A^T) is positive definite. b) Let H be a real, symmetric nxn matrix. Show that H is positive definite if and only if its eigenvalues are positive.

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A)...

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A) > 0 then A is positive definite. (trace of a matrix is sum of all diagonal entires.)

1. Find all eigenvalues of each of the following matrices. (a) |10 −18 b) |2 −1...

1. Find all eigenvalues of each of the following matrices. (a) |10 −18 b) |2 −1 (c) |5 4 2 6 −11 | 5 -2| 4 5 2 2 2 2|

Show: all SU(2) matrices satisfy the 4 defining properties of a group.

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) +...

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) + AB + BA + B^(2). Hint: Write out 2x2 matrices for A and B using variable entries in each location and perform the operations on either side of the equation to determine whether the two sides are equivalent.

Question