3 2 -1 1 Determine whether AT B is invertible Given the matrices A = [...
determine whether the given set of invertible n × n matrices with real number entries is a subgroup of GL(n, R).... The set of all n × n invertible symmetric matrices. That is, the set of all matrices where A^T = A and det(A) notequal 0. [Important things to note are that (AB) ^T= (B^T)(A^T) and (A^T ) ^−1 = (A^−1 ) ^T .]
- In each part, determine whether the pairing (, ) determines an inner product on the vector space V. Justify your answer. (1) V=R>, <[ 3 ][ ==x". (ii) V = R", (7,w) = tr((AU) Aw) where t denotes transpose and A is an invertible matrix. (iii) V =R”, (7,w) = tr((AT)* Aw) where t denotes transpose and A is a non-invertible matrix. (iv) V = C([0,1],R), (f(x),g(x)) = 5o+ f(x)g(x)dx.
(b) In each case below, state whether the statement is true or false. Justify your answer in each case. (i) A+B is an invertible 2×2 matrix for all invertible 2×2 matrices A, B. [4 marks] (ii) If A is an n×n invertible matrix and AB is an n×n invertible matrix, then B is an n × n invertible matrix, for all natural numbers n. [4 marks] (iii) det(A) = 1 for all invertible matrices A that satisfy A = A2....
1. Determine which of the following matrices are invertible. Use the Invertible Matrix Theorem (or other theorems) to justify why each matrix is invertible or not. Try to do as few computations as possible. (2) | 5 77 (a) 1-3 -6] [ 3 0 0 1 (c) -3 -4 0 | 8 5 -3 [ 30-37 (e) 2 0 4 [107] F-5 1 47 (d) 0 0 0 [1 4 9] ſi -3 -67 (d) 0 4 3 1-3 6...
Q1 (9 points) Determine whether the following matrices are invertible and find the inverse, if it exists 2 0 -1 -12 3 -4 -6 8 b) c) 3 2 2 3 0 2 3 1
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
[B0 5. Given that B and D are invertible matrices of orders n and p respectively, and A [W X Find A- by writing A-1 as a suitably partitioned matrix Y Z
Гв C D 5. Given that B and D are invertible matrices of orders n and p respectively, and A = (W x] Find A-? by writing A-1 as a suitably partitioned matrix LY Z
3. optel Let 2 2 2 4 1 3 3 1 1 (a) Determine whether the product AB is defined. If yes, determine whether it is invertible. (b) Determine whether the product ABC is defined. If yes, determine whether it is invertible [epta Consider the system of ne linear equations in n unknowns given by As - b, where A € Mn,n is the n x n matrix of coefficients, as is the vector of unknowns and b E R"...