Let A be a square matrix of size n whose entries are real or complex numbers. Suppose that the diagonal entires of A are much larger than the size of the other entires in the same row, in the sense that, for each row i, n 2||||||aj|| = · Σ ||aij|| = ||ª₁|| + · · · · j=1 Prove that A is invertible. •+||ai|| + · +||ain||.

Q: 9. True or False (T/F): If True, Prove it; If False, give a counter-example. a) For all nonzero nxn…

A:

Q: 1) Show that the real matrix [[0 1 1], [1 0 3], [4 -3 8]] is invertible and determine the inverse…

A:

Q: 2. Express each of the following quadratic forms as a matrix product involving a symmetric…

A:

Q: Let A be a matrix whose entries are integers, and det A=1. 4 : Show that the entries of A-1 are all…

A: given ,det A = 1

Q: Let A be an invertible 3 by 3 matrix. Assume that PA²P-15ATA². = Then det(A) =

A: Step 1:   Step 2: Step 3: Step 4:

Q: Given that A, B, C are all square matrices of the same size, matrices A and C are both orthonormal,…

A:

Q: Let A, B, C be n × n real matrices such that A = BC. Prove A is invertible if and only if B, C are.

A:

Q: If you know all 16 cofactors of a 4 by 4 invertible matrix A, how would you find A?

A: Given, an invertible square matrix A of order 4, and we know all the 16 cofactors of this matrix.…

Q: .

A:

Q: If A and B are invertible matrices, show that AB and BA are similar.

A:

Q: 2. Let a, b be non-zero real numbers. Let T be an n × n matrix. If T satisfies (Tal₁)(T - bln) = Onn…

A:

Q: Let A, B and C be square invertible matrices of the same size. If BTB=I=BBT(XT is the transpose of…

A: A square matrix is said to be invertible if their inverse exists that is determinant of matrix is…

Q: Suppose that the square matrix A is invertible. Show that the inverse of A is unique.

A: See below the explanation

Q: Write f= x + 10x1x2 + x3 as a difference of squares, and f= x? + 10x,x2 + 30x as a sum of squares.…

A: Given f=x12+10x1x2+x22.  Let the associated matrix of f be A=abba is 2×2 symmetric matrix. Then,…

Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

A:

Q: Prove: For all m,n E N, (A†)™ = (A")† holds for all m x n matrices A.

A: Given that m,n∈ℕ. Let A be an m×n matrix. Let A+=Z. We know that A+=Z is pseudo inverse of A if it…

Q: Let x = (−3, 1, 7, 2, 3, − 1)T, find P, Householder matrix, and a scalar a, such that Px = y, where…

A: A Householder matrix is a type of orthogonal matrix that is used to perform Householder…

Q: o matrices A and B one has that Null(B) is contained in Null(A · B).

A: Given, for two matrices A and B, to show that Null (B) contained in Null(A·B).

Q: Let A and B be unitarily equivalent complex matrices. Show that A is normal if and only if B is…

A:

Q: Find an example of a 2 x 2 matrix A such that A? + I2, A3 + I2, and A6 = I2. Explain how you know,…

A:

Q: 3. Prove that (i) /, =1, and (ii) ||, =Vn (I is the Identity Matrix of order n).

A:

Q: 2. a c) Find all the matrices A = [] with real entries such that A² = [ 2] detail work. Show your

A:

Q: 2. (i) A square matrix A is called idempotent if A? = A. Find all 2 x 2 matrices A over the real…

A:

Q: Write down a n × n matrix A such that Cond(A) =1, but determinant(A) → 0 as n → ∞.

A: claim- write down a n×n matrix A such that CondA=1, but detA→0 as n→∞. it is known that condition…

Q: A transformation is defined by the 2 × 2 matrix A =  −a b − a a + b a  where a and b are scalars.…

A: please see the next step for solution

Q: Let A be a square matrix of size n whose entries are real or complex numbers. Suppose that the…

A:

Q: 7. Prove that ||A|| = omax (A) for a real matrix A.

A: For euclidean norm the norm of matrix A may be defined as follows ∥A∥=sup{⟨Ax,y⟩   :x,y∈Rn,…

Q: If A is invertible, which properties of A remain true for A-¹? (a) A is triangular. (b) A is…

A: 1. The inverse of a triangular matrix is triangular. So this property remains true.2. The inverse of…

Q: Show that whether an m xn-dimensional matrix as is a unary function when m <n for vector space R

A: We have to show that an  - dimensional matrix as is a unary function when  for vector space…