The Cayley-Hamilton theorem from Linear Algebra states that if A is an n x n matrix with characteristic polynomial p(λ-det(A-AI) then p(A) 0, where the last 0 is the n × n zero matrix. Verify that Cayley-Hamilton holds for the matrix 2 -2 0 A 0 2 0 -2 2 2

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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.

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Q: Consider the 6 × 8 matrix 3 0 0 0 0 1 2 1 0 0 0 0 1 0 0 -3 4 -7 -9 A = 0 0 0 0 0 0 0 0 0 1 8 1 and…

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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: 2) An n x n matrix A is called nilpotent if Ak = 0 for some positive integer k. Show that if A + 0…

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Q: 3. (a) If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

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Q: 7. Matrix A is said to be involutory if A² = I. Prove that a square matrix A is both orthogonal and…

A: In this question we have to prove that a square matrix A is both orthogonal and involuntary.

Q: Find a matrix A such that [is: is a solution for AX=X'.

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Q: 8. Let X be an n x p matrix of full column rank. (a) Use Theorem A.23 to show that (In + XX')- = In…

A: X be an n × p matrix.(a) Let A=In, B=X, C=Ip, D=X'Then (A+BCD)-1=(In+X Ip,X'-1=(In+XX')-1 and…

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Q: The Cayley-Hamilton Theorem states that a matrix A satisfies its own char- acteristic equation. That…

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Q: .1 Suppose A is a matrix with characteristic polynomial CA(x) = x²(x – 1)³(x+2)². Determine the…

A: The solution are next step

Q: 4) Confirm the results of the Fundamental Theorem of Linear Algebra for each of the matrices A and B…

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Q: A Hessenberg matrix H, is n x n matrix that has 2 on the main diaonal, 1 everywhere below the main…

A: As per the instruction, we will be answering only the first one. For (a), det(H2)=det2112…

Q: Show that the characteristic polynomial of A is p(λ)=λ2−5λ+6.

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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…

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A: Given, matrix A is diagonalizableand characteristics polynomial of Ap(λ)=-λ2(λ+3)We can find…

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A: Given matrix is A = -5-2005-1-2-30

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Q: Refer to the following theorem: If A is an mxn matrix with linearly independent columns, then A can…

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A: Solution (a):- Given Characteristic polynomial of matrix A =  (2-x)(x2+1-a2) But we have to find the…

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Q: PROVE IT Given any real n x n matrix A, if A is invertible, then there is an orthogonal matrix Q…

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A: We shall solve first question only with both the subparts . For others kindly post again by…

Q: If A? is a zero matrix, then AT A = 0 is: %3D O a. Not possible unless A is a zero matrix O b. not…

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Q: snip

A: Givenp(x)=xn+an-1xn-1+...+a1x+a0Also Companion matrix.

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A: Please see below picture for detailed solution.

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Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.