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Question: determine if each of the following statements is true or...

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Determine if each of the following statements is true or false. Justify your answer by a proof or a counter example. Proofs should be based on your linear algebra background or the material provided in my lecture note. 1. The equality хту-утх holds for any two n-dimensional column vectors x and y. 2. If A, B and C are invertible matrices of the same dimensions, then the product of their transposes, ATBTCT, is invertible. 3. If A, Band C are invertible matrices of the same dimensions, then their sum AB +C is invertible. 4. Suppose that A is an invertible matrix. Switch its top two rows to obtain a new matrix B. The matrix B must be invertible as well. 5. Let vi,,2 . . . ,ył be column vectors of the same dimension, and let v-v1 + + VA. The vectors + v!,.. . *.v are linearly dependent. 6. The system Ax- b has a solution, if b is a linear combination of columns of A 7. For a system of linear equations that contains 2 equations and 3 variables (ie, A є R2x3), there are always infinitely many solutions.

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