2. Let (r x s) denote an r x s matrix. Find the size of those matrix products that defined. a) (2 x 3)(3 x 4) b) (1 x 2)(3 x 1) c) (4 x 4)(3 x 3) d) (4 x 1)(1 x 2)

Q: Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric,…

A: According to the given information suppose, It is required to show that if A is symmetric then R is…

Q: Find a 2 x 2 matrix A for which {;}. 5 E1 = span = span

A:

Q: • The Matrix B from 3 x 2 at by = 1- is 1. 3. 4. 1 -2

A:

Q: Compute the product 0 () 0 (²) as a 2 x 2 matrix. What kind of isometry is this product?

A: Given product is  Oπ5O2π15 where Oθ represents rotation matrix through an angle θ in anti-clockwise…

Q: Verify that if r < s, then rows rand s of a matrix can be interchanged by performing 2(s - r) - 1…

A:

Q: z= 5x, + 12x2 + 4 x, S.t X, + 2x2 + X, s 8 2 x, - X2 + Xạ = 8 where x,, X2, Xs 20

A: For a Minimization Problem, The constraints must be in the form≥ in the standard Primal Form.…

Q: CD, where C and D are n x n matrices and C is invertible. Prove that the matrix product DC is…

A:

Q: Let A be square matrix. If A" = 0. A= Oava %3D for n> 2. Then

A:

Q: How is the matrix of R^−1 obtained from that of R in general, that is, when A = {a1, . . . , am}, B…

A:

Q: 3. Let A be an m x n matrix and be Rm. Given that Ax = b has two distinct solutions u and v. Explain…

A:

Q: Give an example of a 2 x 2 matrix A such that im(A) =ker(A). %3D

A: Null space of a linear transformation: T:V(F)→V'(F) be a linear transformation then the null space…

Q: e/ By three ways Show that B= %3D isorthogunal matrix

A:

Q: 2. (a) Express -2+t+t +r' as a linear combination of 1-t, 1+r, and 1+1- or show that it cannot be…

A: We have to write in the linear combination.

Q: Let A 1B = 5 l, and C-1 = If X is a matrix that satisfies: %3D %3D 2B = Al2 - XC then X is

A:

Q: B = . and = If X is a matrix that Let A = satisfies: Al, = 2B – XC. Then X is %3D

A:

Q: Suppose (B-C)D=0, where B and C are mxn matrices and D is invertible. Show that B = C.

A: The given matrix equation is B-CD=0 B and C are the matrices of the order m×n while D is the…

Q: .

A:

Q: [:] Suppose A is a 2×3 matrix such that A and A -3 Compute the product A

A: I have writen the vectors in linear combination of given two vectors and applied A in both sides

Q: Suppose A is a 2 x 2 matrix such that A and 1 A What is A-1?

A:

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: How many rows and columns must a matrix A have in order to define a mapping from R* into R³ by the…

A: Find the number of Rows and columns in a matrix A.

Q: 4. Decide if there is a three by three matrix A such that A. 1 for all æ, y, z in R.

A:

Q: Let A and B be n × n matrices such that AB is invertible. Show that A must be invertible. You cannot…

A:

Q: Find a 3 x 1 matrix x with entries not all zero such that 1 2 -1 Ax = 1x, where A 1 1 -4

A: A=1  2-11  0  14-4  5 Now let x=abc be the required matrix. Now consider, Ax=1x⇒1  2-11  0  14-4…

Q: Show that if annxn matrix U satisfies (Ux)· (Uy) = x•y for all x and y in Rn, then U is an…

A:

Q: The Matrix B from 3 x 2 at b = 1- - is %3D 1. 2. 3.

A:

Q: Suppose that A is a 5 x 5 matrix which has a null space of dimension 4. The rank of A is rank(A) =

A:

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: How many rows and columns must a matrix have in order to define a mapping from R5 into R7 by the…

A:

Q: Show that ∣A∣ = ∣AT∣ for the matrix below.

A: Now the transpose is,

Q: 2 - matrix A and that h(T) = T? – 2T – 7 -

A:

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

A:

Q: Find the value of x so that the matrix is not invertible. 1. 3 2 0 [ 1 2 1 If answer is in decimal,…

A:

Q: 13. Find all 2 x 2 matrices A = (" ta² = (C :) such that %3D

A: We need to find the set of all 2×2 matrices such that A2=O, where O is the zero matrix of order 2.

Q: 2. Verify Calay-Hamilton theorem for the following matrix: (2 1 -1 A = (3 1 1 0 2

A: Solution:Given        A=21-1310102To findA-λI=0

Q: Let A = CD, where C and D are n × n matrices and C is invertible. Prove that the matrix product DC…

A: A = CD, where C and D are n × n matrices, and C is invertible. We need to Prove that the matrix…

Q: Suppose AB = C, where A and C are upper triangular. (a) Find 2 × 2 nonzero matrices A, B, C, where B…

A: Answer

Q: 1. For any 3x3 matrices A & B: Verify that (AB)" = B"A". %3D

A: By using the properties

Q: -) Let A, B be nx n symmetric matrices such that XTAX = X™BX for all %3D column vectors X e R". Show…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in a line and connected by…

A: (a) The 4 x 4 matrix equation relating the column vectors f and x will be:

Q: 2. Write the quadratic function f(x,,y;)=-3x; +2x,x, – 3x; – 4x, – 4x, +6 in matrix form as f(x) =…

A:

Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

A:

Q: Let A = (a;;)ij be an m x n matrix of real numbers, and let f : R" Rm be defined by 0-0 x2 x2 f = A

A: To find the derivative of f given: matrix A which has m rows and n columns  also the function f…

Q: Theorem: For AC Cmxn, R(A)IN(A*)

A: We need to show that R(A) is perpendicular to N(A*).

Q: If A is a skew-symmetric matrix then show that AAT = AT A and A² is symmetric. [Note AT denotes the…

A: We will use definition

Q: Let A be square matrix. I f A"= Onwa for %3D n> 2. Then A= Onun

A: we will see example and conclude.

Q: 2. Find the matrices P and D for an orthogonal diagonalization A=PDPT for 3 0 0 A=

A: First find eigenvalues eigenvactors corresponding to eigenvalues then we get matrices P and D.

Q: Write down what it means for a n x k matrix to be singular.

A: As we know, If matrix is singular then it has determinant equals to '0' For Finding determinant…

Q: For A e Mnxn(C), the function ||A||. = max1<ijśn |aij| is a norm on Mnxn(C) but not a matrix norm.…

A: Matrix Norm: Let Mn×n(ℂ) be the vector space of all matrices of order n with entries in the field ℂ,…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.