Question: Let A be an n n matrix. Prove that if Ax = x for all n 1 matrices x, then A = In.

Let A be an n × n matrix. Prove that if Ax = x for all n × 1 matrices x, then A = In.

Step by Step Solution

★★★★★

3.49 Rating (152 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

If Ax x for all n 1 matrices X t... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

951-M-L-A-L-S (6826).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Let A be an n n matrix with singular value vector = (1,... , 1). Prove that (a) |||| = ||A||2; (b) ||||2 = ||A||F, the Frobenius norm of Exercise 10.3.16. Remark: ||||2 (also defines a useful...

Q:

Let A be an n n matrix and F be the function from Rn to Rn defined by F(x) = Ax. Use the result in Exercise 12 to show that F is continuous on Rn.

Q:

Let A be an n n matrix and a scalar. Show that det(A) = n det(A)

Q:

2. Show that the eigenvalues of a triangular matrix are the diagonal elements of the matrix. 3. Let A be an n x / matrix. Prove that A is singular if and only if ) = 0 is an eigenvalue of A. 4. Let A...

Q:

Let A be an n n matrix. Prove that the following statements are equivalent. (a) N(A) = {0}. (b) A is nonsingular. (c) For each b Rn, the system Ax = b has a unique solution.

Q:

Attached is the question Problem 3. (20 points.) Suppose A is an m x n (constant) matrix, B is a p x q (constant) matrices, X is an n x p random matrix, Y is an n-dimensional random column vector,...

Q:

Attached is the question Problem 3. (20 points.) Suppose A is an m x n (constant) matrix, B is a p x q (constant) matrices, X is an n x p random matrix, Y is an n-dimensional random column vector,...

Q:

Let A be an invertible n n matrix, and let B be an n k matrix. Prove that if the columns of AB span R^n , then the columns of B also span R^n

Q:

Let A be an n n matrix. Prove that the characteristic polynomial of A has degree n. Prove that the coefficient of n in the characteristic polynomial is 1.

Q:

Let A be an n n matrix. Prove that A is non singular if and only if the columns of A span F n .You may use the most recent version of the nonsingularity theorem in your proof

Q:

Which of the following circuit does not satisfy the Boolean expression AB + AB = F A TO (a) B (c) F A B A (b) B (d) B PD F A D F

Q:

II. VK provides the following information for the year ended March 31, 2019. Opening balance of provision for bad debts Rs. 12,450. i. Opening balance of provision for discount on debtors Rs.4,500....

Q:

The United States is trading salmon to Peru in exchange for anchovies. If these nations are trading based upon relative opportunity costs, what must be the case? (A) The United States has comparative...

Q:

You are a project manager for your organization. Your project is to install several devices for one of your companys clients. The client has requested that you complete a few small tasks that are not...

Q:

Consider the space of all functions of the form a + bt + cet + de-t, where a, b, c, d are scalars. (a) What function in the space corresponds to the sum of (1, 2, 3, 4) and (- 1, - 2, 0, - 1),...

Q:

The set of all real-valued functions f defined everywhere on the real line and such that f (1) = 0, with the operations defined in Example 4. In Exercise a set of objects is given, together with...

Q:

The set of all positive real numbers with the operations x + y = xy and kx = xk In Exercise a set of objects is given, together with operations of addition and scalar multiplication. Determine which...

Q:

Everything I have is correct, but I cannot figure out how to get Y for both standard deviations and the variances. can someone show me how to get Y for both. Please and Thanks in advance Use the...

Q:

A portfolio has three stocks - 240 shares of Yahoo (YHOO), 110 shares of General Motors (GM), and 30 shares of Standard and Poor's Index Fund (SPY). If the price of YHOO is $30, the price of GM is...

Q:

Forecasted Statements and Ratios Upton Computers makes bulk purchases of small computers, stocks them in conveniently located warehouses, ships them to its chain of retail stores, and has a staff to...