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: 2. Find a vector orthogonal to the nullspace of the matrix [1 27 6. 4 L5 -1 2.

A: According to the given information it is required to find a vector orthogonal to the null space of 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: Let W be the set of all vectors of the form 4r3s+t 2s 2r t - - 4s 2r 3t - - - (r+5s) with r, s and t…

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: Q5. Show that the function from R"xn to R given by ||4||max = max |a¡j| i,j is not a norm by stating…

A: Given: A function from ·max : ℝn×n→ℝ defined by, Amax = maxi, jaij

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: 04) X = [ Y = ; Hand Z = [ 3 Q.4) X Prove the Distributive laws of multiplication over Subtraction…

A: The given matrices are X=23-10,Y=0131  and  Z=22-10. The distributive law of multiplication over…

Q: find the twelfth pawer X of the matrix X that satusfies the Equation 12 X A = AD

A: XA=AD⇒X=ADA-1⇒X12=ADA-112=ADA-1·ADA-1·…·ADA-1=ADA-1ADA-1A·…·A-1ADA-1=AD12A-1

Q: Let A be a 3 x 4 matrix such that solutions to A = 0 are linear combinations of X and X2 vhere Find…

A:

Q: 3 21 3. The all values of a and B, so that the matrix 4-3 1 4 is positive definite, are [2 B 2] (b)…

A:

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: solve the given initial-value problem Q. y′′+2y′+5y =4sint +δ(t −π/6), y(0)=0, y′(0)=1.

A:

Q: Find the elgenvalues of the matrix [ 2 1 1

A: We have to find the eigen value of the given matrix.

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

A:

Q: Show that the function from Rnxn to R given by ||A||max = max |aj| is not a norm by stating which of…

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: 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: 4. In fitting a straight line, the formula to find the sum of the squares of the residuals is (a) E…

A: 4. Hint: Here we need to just write the sum of the squares of the residuals.

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: 1. Show that the matrix 2 1 A = 1 2 -1 -1 is positive definite.

A: Given :- a matrix A  We need to show that a matrix is positive definite. We know that , A symmetric…

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 Less positive Consider a symmetric positive definite matrix A separated into subblocks according…

A:

Q: ion of the IVP wrong and why? Sy' = 3y²/3 y(1) = 0.

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 M (for matrices) be the set of all 2 x 2 matrices over the reals and x, y ∈AM. Does…

A: This statement is not true.

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

A:

Q: 2 Be A, a square matrix i. Given that A holds = 44 + 5I . Prove that A is reversible, and express A-…

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

A:

Q: PS1(2) If A, B = Mn(F) and a, BE F, prove that tr (aA + BB) = a tr A + 3 tr B.

A: Prove that if A, B∈MnF and α, β∈F, then trαA+βB=α trA+β trB.

Q: What is the null space of this matrix:   A = 3 0 0 2 0 0 1 1 1

A:

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.

Q: Let M = -3 Find c and c2 such that M² +cqM+c»I½ =0, where I, is the identity 2 × 2 matrix and 0 is…

A:

Q: RCISE 1. Given A = 2 E , verify that (a) (A+ B) +C = A+(8 +C) (b) (A + B) – C = A+(B – C)

A: topic - matrices

Q: A is an n x n matrix. Check the true statements below: A. Col(A) is the set of all vectors that can…

A: