True or False? If False, explain why. 6. For any invertible matrix A, the matrix (AA^T) is positive definite. 7. If A is an orthogonal matrix, then det A= 1. 8. Two matrices are similar if and only if they have the same characteristic polynomials. 9. Any finite dimensional inner product space has an orthonormal basis. Solution 6) we know that : A real symmetric matrix \'M\' is positive definite iff there exists a real nonsingular matrix \'A\'such that M=AAT so (6) is true. 7) AAT=I => |AAt|=|A|2=1 => |A|= 1 or -1 so false 8) Similar matrices represent the same linear transformation so TRUE 9) TRUE : it follows matrix theorem refer http://cims.nyu.edu/~kiryl/Algebra/Section_4.4--Inner_Product_Spaces.pdf (theorem 4.4.2) 10) FALSE, innet product is defined by XTY only.