2. Consider the matrix a A = 1 11 1 a 1 a 5, where a is some real number. Use the criterion based on principal minors discussed in lecture to show that A is positive definite if and only if 1 < a < 2.

Q: 3| |1 Let A 3 1 4 3 1 3 41 1) Verify that: AadjA = det(A) I (Where I is the identity matrix) 2) Find…

A: given matrix isA=1331431341) to verifyA.adj(A)=det(A).I2) to find A-1

Q: Assume that A, B are square matrices and that AB is invertible. If AB = I, show that B is…

A:

Q: 5. Let S = {V₁, V2,..., Un}, n ≥ 2 be a set of nonzero vectors of V. Each of the following…

A:

Q: 2. Let n x n matrix A have the form Au Azz A = where An is k x k (where k < n). Show that A is…

A: The given matrix is  A1100A22 A matrix is called non-sigular if the value of its determinant is…

Q: [1 A = 7 3 9 13 15 5 11 17]

A:

Q: 1. Let A be a 6 x 14 matrix (6 rows and 14 columns). Show that the null space of A has dimension at…

A: Let A be an m × n matrix. rank(A) ≤ m and rank(A) ≤ n for any m × n matrix A. Here, A is a 6× 14…

Q: Let A be an mxn Aqtrix Prove that = max |Ax/ -max E l9;

A:

Q: Let A be an 2 x 3 matrix and b be a vector in R? such that the system Ax = b has solution set 0. +t…

A:

Q: 5. Find the three by three matrix A such that [r – y] A. y for all r, y, z in R.

A: In the given problem we have to find the three by three matrix A such that…

Q: Prove the distributive property of matrix multiplication over vector sums for a given matrix A E…

A:

Q: Let (A.x, Ay, As) solve AT I JI Ax Ay I.l - S 0 X As Assume ri + 0, S and X are positive definite…

A: As per the question we are given a linear system. From which we have to find ∆x , ∆y , ∆s And prove…

Q: Indicate whether each of the following statements is True (T), or False (F). Explain your answers. 1…

A: See the solution in the given fig.1

Q: 4. Prove: If b is a non-zero vector, and x and x are both solutions to the matrix equation AX =6,…

A:

Q: Let A E Cmxn and be Cm. Show that the following three statements are equivalent: . There exists a…

A: As per the question we have a matrix A ∈ Cm×n and a vector b ∈ Cm  Now we are given the following…

Q: Let A be an n ×n matrix. Prove that dim(span({In,A,A2, . . .})) ≤n.

A:

Q: Let A, B; be Square matrixes, which meet the following conditionS BA = A(A+ I) & A3 I %3D 1. Prove…

A: We will use properties of matrix multiplication

Q: 3. Find all real numbers x, y such that the columns of the following matrix are linearly de-…

A:

Q: Let A E Cmxn and be Cm. Show that the following three statements are equivalent: . There exists a…

A: As per the question we are given 3 different statements about the matrix A ∈ Cm×n and the vector b ∈…

Q: Consider the following statements: 1 4 -6 0 0 9 1 7 I. The matrix is stepwise. II. If A is a square…

A:

Q: Which of the following matrices is a nilpotent matrix of degree 2. a a) where ab =-x². -x a where ab…

A: We have to select the nilpotent matrices of order 2 from the above list.

Q: Let Q be a orthogonal matrix Prove det Q = ± 1

A: det Q = ± 1, Q is an orthogonal matrix.

Q: 20) Find the conditions on a and d such that the matrix A = [] has and inverse. What is the inverse…

A: Given,

Q: 3 -1 Let A = and B = 1. %3D a) Find all IR Such that the in Columns of AB lin early dependent are to…

A:

Q: 1.2. Prove that for every matrix A, (AT) = A

A: Introduction: There are many properties that are exhibited in a matrix. One such significant matrix…

Q: 1- nullspace af now equivalent to the for the 1 1- 2. C. that puy -2 B. assume matrix pand Let A

A:

Q: Let A and B be two square matrices of order b x n. Prove that A and B are non singular if and only…

A:

Q: 7. Let A be an 7 × 7 matrix in which every entry is 1 and I, be the 7 × 7 identity matrix. Prove…

A:

Q: Consider the following block matrix: N a aT c where N is a m ×m real symmetric matrix, a is a…

A: We will use the basic knowledge of matrices to answer this question.

Q: (3) Either prove or provide a counterexample for each of the following statements. I. All scalar…

A:

Q: [: ] 4. Let A Show Ais the product o f elementary 2 4 matrices

A:

Q: 3. Matrix Range, Rank, and Pseudo-inverse Let A e Cmxn and b E C. Show that the following three…

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: A is a 9 x 4 matrix. Which of the following could be nullity(A)+nullty(Atr). a) 3 b) 4 c) 6 d) 7 e)…

A: Concept:

Q: Is the vector [1/2] 1 in the null space of matrix A 8 10 L22 7 3 -1 13 -6? 2

A: Let us check whether the given vector is in the null space of given matrix A.

Q: 5. For each of the statements given below decide if it is true or false. If it is true explain why.…

A: We check that given statement is true or false.