Consider M as an n x n matrix, we can say that M is nilpotent  if Mx = 0 for some x ∈ +Z . Now: a.) Prove that every nilpotent matrix is singular and verify if  [ 0 1 1   0 0 1   0 0 0 ]  is nilpotent.  b). If M is said to be nilpotent, then prove that In- M is nonsingular

Q: Prove that for an n×n matrix A, if A^2 is invertible, then A is invertible.

A: Since, if a matrix A is non-zero then A is invertible.

Q: Prove that if one row (column) of the n × n matrix A consists entirely of zeros, then A is singular.

A: We know that if all entries of one row(column) of the n×n is zero then determinant of matrix is…

Q: 2. Regarding DFT matrix W₁, prove that W 1 = W for N=4. N

A: Consider the given equation, WN-1=1NWNH for N=4 Where, WN is a DFT matrix. So, as we know that,…

Q: Prove or disprove: For any n×n matrix A, AT A = AAT

A: For any n×n matrix A, ATA=AAT

Q: Prove that if B is a 3×1 matrix and C is a 1×3 matrix, then the 3×3 matrix BC has rank at most 1.…

A: Let A be the 3 × 3 matrix of rank 1, then it will be in the form as follows.

Q: Let A be any m×n matrix, Omn be the m×n zero matrix, and c be scalar. Show that if cA = Omn then…

A: Let A be matrix of order m× n Then A = aij   where i is row and j is column ,   i = 1, 2, . . . m…

Q: Let S be an n × n singular matrix. Prove that for any n × n matrix B, the matrix SB is also…

A: Singular matrix :A matrix is singular if and only if it has a determinant of 0

Q: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: Let A be an n×n matrix. Prove that A and AT have the same Jordan canonical form.

A:

Q: Prove that for any m×n matrix A, rank(A) = 0 if and only if A is the zero matrix.

A: Given:Prove that for any m×n matrix A, rank(A) = 0 if and only if A is the zero matrix.

Q: if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))

A:

Q: Let A be an m × n matrix. Prove that N(A) ⊂ N(ATA).

A:

Q: Show that if A is any mxn matrix, then Im= A.

A:

Q: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

A: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: Consider the matrix A E RmXn with m >n and linearly independent columns. Also consider M = A(AT A)-'…

A:

Q: Prove that if A is an n X n matrix, then A - Ar is skew-symmetric

A:

Q: Let C be a nonsymmetric n × n matrix. For each ofthe following, determine whether the given…

A: Given  B=C-CT Find transpose.

Q: Let A be a nonsingular n×n matrix with a nonzero cofactor Ann, and set c = det(A)/Ann Show that if…

A: Let A is any non singular nxn matrix with non zeroCofactor Ann and Set C = det(A)Ann   By concept…

Q: Prove that if A is an invertible matrix and B is row equiva- to A, then B is also invertible. ation

A:

Q: Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗A is a diagonal…

A: Given that A is an m*n matrix in which no two column are identical.Since A*A is defined only if A…

Q: Let A be a m × n matrix with columns c1, …, cn. Prove that {c1, …, cn} is independent if and only if…

A:

Q: Prove that an upper triangular n ×n matrix is invertible if and only if all its diagonal entries are…

A:

Q: Prove If U is a unit upper triangular matrix, then U is nonsingular and U−1 is also unit upper…

A:

Q: If Q+A= A holds for every m×n matrix A, show that Q = 0mn.

A: Consider the given equation, Q=A+A   Now, subtract the matrix A from both side: Q+A-A = A -A As, A…

Q: Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B =…

A:

Q: .

A:

Q: Find a 3 x 3 matrix A such that Az = 10 for all i in R³. A =

A: Given Ax=10x  We have to find 3×3 matrix A  Now

Q: Suppose A is an eigenvalue of the n x n matrix A. If u and v are eigenvectors of A corresponding to…

A: Given: Suppose λ is an eivenvalue of the  n×n matrix A and u, v are eigenvectors of A corresponding…

Q: Let A be an m × n matrix. Prove that if AT A is nonsingu- lar, then rank A = n.

A: Given that, A is an n x n matrix. A^T is the transpose matrix of A. AT A is non-singular. We have to…

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D…

A: Given that ,  A       magic square  To show that : We can write as                     Where,  is a…

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: Show that if A is any m × n matrix, then Im A = A and A In = A.

A: Given: Matrix A of order m × n.

Q: Let A be a m x n matrix, such that Ax = 0 has parametric solutions of the form x(s, t, u) = sX1 +tx2…

A: Please check step 2 for the solution.!

Q: Construct a 4 x 3 matrix A = [ āj az āz ] such that {ā1, ā2 } is a linearly independent set and az…

A:

Q: Explain why the columns of an nxn matrix A span R" when A is invertible. Choose the correct answer…

A:

Q: Suppose V and W are finite-dimensional and T E L(V,W). Show that with respect to each choice of…

A: Suppose V and W are finite-dimensional and T∈LV,W. To show: to each choice of bases of V and W, the…

Q: Let A be an m × n matrix. Show that ATA and AAT are both symmetric.

A: Given A be an m×n matrix. we have to show that ATA and AAT are both symmetric.we will make use of…

Q: Let B be an n x m matrix with rank m. Prove that there exists an m x n matrix A such that AB = Im.

A: Consider the given matrix n×m with rank m

Q: Let An be the n × n matrix given by If ∆n = det An, prove that ∆5 - 2 cos θ∆4 + ∆3 = 0

A:

Q: Suppose that a matrix M is nilpotent if exists an x element of Z+ in where Mx = 0. If M is…

A: We will prove the given result.

Q: Let A be a 4x4 singular matrix. What is the smallest possible dimension of Null Space of A? A) 2

A: Using,  Rank- Nullity Thereom

Q: Let A be an m x n matrix. Prove that if B can be obtained from A by anelementary row [column]…

A:

Q: If A is an n x n symmetric matrix and B is any n x n invertible matrix, prove that A is positive…

A: Consider the given conditions. The A = n × n symmetric matrix and B = n × n invertible matrix.…

Q: A

A: