4. Determine all values of p such that the matrix A is invertible. 1 р 2 A = р 3 4 1 1 -2

Q: t A and B be arbitary n x n matrices. If B is invertible, prove that AB- = B-'A if and only if AB =…

A:

Q: a) If a 7 × 5 Matrix A has rank 2, find dim(Null A), dim (Row A) and rank AT

A: 1) Given  7×5 matrix  rank(A)=2 using rank nullity theorem we have rank A + null A= number of…

Q: 1. Find the cofgctor of 6 in matrix B. 4 り り

A: We will find out the cofactor of 6 for given matrix B.

Q: 1 Suppose that a 3 x 3 matrix A has det A = 1 and that its cofactor matrix is C = 1 2 |-7 4 2 Find…

A: Formula: Let A be a n×n matrix with non zero determinant then, A-1=1det ACT where C is the cofactor…

Q: 1) Construct the matrix A =(a;;) where 3x2 ) aj = i+j, if i = j,and Li-j, if itj %3D

A:

Q: 12 Let A be a square non-invertible matrix. Which of the following is not true? Select one…

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Q: For which value of x is the following matrix not invertible: 1 1 3x A = 3x 3x 3x 3x 3x O 2 4 O1/3 1

A:

Q: 3. Let A be a m X n matrix with columns C1, ... , Cn . Prove that {c1, ..., C,} is independent if…

A: In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a…

Q: (a) A square matrix M satisfies M = -M. Show that I- M is always invertible.

A:

Q: Consider the Az 1 34 17 9 203 2 8 16 6 0 5 /

A: (a) We know that we can find the determinant of the matrix by expanding along any column or row.…

Q: 6. Suppose  X + XB = C, where X, B, and C are square matrices of the same size. a.  Derive an…

A: We will see that (I+B) must be invertible for existence of X.

Q: a If A the matrix A is invertible if and only %3D lc d]° if

A: Find the determinant of the matrix A. A=abcdA=a×d−b×c=ad-bc

Q: 2. (a) Suppose (B − C)D = 0, where B and C are m x n matrices and D is invertible. Show that B = C.…

A: Explanation of the answer is as follows

Q: 1.Find matrix p that orthogonally diagonalizes A, and determine p-én AP. Given A=[2 12

A:

Q: 7 Show that A = 13 and B = 4 2 are similar matrices by finding an invertible matrix P satisfying 2 A…

A: Given,A=7-132-4 abd B=1-2-22Let,P=abcd , and P=ad-bc=nSo, P-1=1nd-b-ca where n=ad-bc=dn-bn-cnan

Q: (4) Given a matrix X of independent variables, psudo-inverse X† is defined as (X"X)-'x" (a) If X is…

A: If a matrix is of order (mXn) then transpose of A is of order (nXm).

Q: 5. Suppose that A is a 3 x 5 matrix, and the rows of A are linearly independent. What is the…

A: A is a 3x5 matrix. And rows of A are linearly independent. So that dimension of range space of Ax…

Q: 4. If the product D = ABC of three square matrices is invertible, then A must be invertible (so are…

A: Given that the product D=ABC of three square matrices is invertible. Also given that A must be…

Q: 4 9- That is, find an -5. | Diagonalize the matrix A [3 invertible matrix P and a diagonal matrix D…

A: Given matrix A=4-63-5

Q: For which value of x is the following matrix not invertible: 1 1 2x A = 1 2x 2x 2x 2x 2x 4 1 -1 1 2

A: For a matrix A to not be invertible: dat(A) = 0.

Q: 4. Prove or Disprove: For anynxn matrix A, A A = AA.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: **32. Suppose A is an m x n matrix and x e R" satisfies (ATA)x = 0. Prove that Ax = 0. (Hint: What…

A: Here we use property of modular and dot product.

Q: 5. Find if possible a nonsingular matrix P such that P'AP is diagonal. [1 2 a. A= b. A= 4 -8 -8

A: Given matrix  a.) A=122-1 b.)A=74-44-8-1-4-8-1 To find possible a non singular matrix P

Q: 3.7. Prove or disprove: If the columns of a square (n x n) matrix A are linearly independent, so are…

A: …

Q: [1 1 Show that the matrix A is not diagonalizable.

A:

Q: 5. Suppose that A is an invertible n ×n matrix with inverse A-!. Show that det (A) is nonzero and…

A:

Q: determined 8. If A is an mxn matrix and AA is invertible then the rank of A is cqual to b. m-n C. d.…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: (a) The matrix ( (AB)-1 )T comes from (A-1 )T and (B-1 )T In what order?(b) If U is upper triangular…

A:

Q: Construct the matrix with rank one that has Av = 12u for v = ½(1, 1, 1, 1) and u= ½(2, 2, 1). Its…

A: Given: Let A be a given matrix. rank (A) = 1 To find: The only one singular value for  Av = 12 u .

Q: a) compute det(A),det (B) b) Is the matrix A invertible? Why?

A:

Q: 2. Find the values of k for which the matrix B is invertible. -2 B= 0 k 1 3 2 1 4 -1

A:

Q: 1. Show that matrix A is singular 1 2 A= 3 -1 -2 3 -2

A: Given  A=1203-12-23-2

Q: 13. a) Show that if A and B are n x n matrices, at least one of which is invertible, then AB and BA…

A: (a) If A and B are two n×n matrices, and one of them is invertible, then show that AB and BA are…

Q: Let 4= [ 12 -0,²5] 3 Find an invertible matrix S and a diagonal matrix D such that S-¹ AS = D.

A:

Q: 3. Show that for any n x n matrix A of rank r, we have ||4|| < ||4||ES VFl|4||.

A: To show that: A≤AF≤ rA  for any n×n matrix A of rank r.

Q: 2 4. Find the matrix A satisfying A and A 1 and write the SVD of A in vector form. [Hint: Start by…

A:

Q: Suppose that A is an invertible matrix with n columns, and entry aij in row i and column j. 1.…

A: Let matrix is A and the entries are aij .

Q: 4. If A is 4x4 normally distributed random matrix and I is 4x4 identity matrix. Proof that: a) A'A =…

A:

Q: 6) Find the SVD of the matrix by manually calculating U 4 11 14 A = 8 7 -2]

A:

Q: 2. Every square matrix A can be decomposed as a product of three matrices A = SDR where S and R are…

A: We shall give an counter example for part (2)

Q: 1. Suppose (B – C) D = 0, where B and C are m x n matrix and D is invertible. Show B = C

A: ( B - C )D = 0 where B and C are m × n matrix and D is invertible.  Then, we have to prove that B =…

Q: If an nxn matrix A is invertible, then the rowa of A' are linearly independent. T O True O False

A: If a matrix is invertible then its rows are linearly independent

Q: (b) Write down an invertible matrix P such that [o 0 -21 P-11 2 li 0 [2 0 0] 1 P = 0 20 lo o 1 3 P=…

A: 1)

Q: 2 Suppose A = is positive semi definite matrix then: 10 O a. a < -V20 O b. a = ±v20 O c. lal < 20 O…

A:

Q: (3) Consider the matrix [1 k 31 A = 2 4 -6 2 For what values of k is A invertible?

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Q: Verify that P = is an orthogonal matrix. 1/32/3 N3

A: Given the matrix P=23-23132313-23132323We need to check whether the given matrix is orthogonal…

Q: 3 5. Suppose that A is an invertible matrix with A-1 a. Find A.

A:

Q: If A is an n×n matrix having linearly independent eigenvectors v1,v2, . . . ,vn, then the n×n matrix…

A: Matrix and whose columns which are linearly independent

Q: 5. Show that the condition that the block matrices (8 ) m ( ) A 0 А С and B В can be matrices of the…

A: Given: The block matrices, A00B and AC0B.   To show: The given block matrices can be matrices of the…