Determine whether each statement is true or false. If it is true, justify it. If

Question

Determine whether each statement is true or false. If it is true, justify it. If it is false, provide a counterexample. a). A linear system with 2 equations and 3 unknowns has either infinitely many or no solutions. b). If A = [v_, v_2, ..., v_n] is an n times n invertible matrix, then it has rank n. c). If A = [v_1, v_2, ..., v_n] is an n times n invertible matrix, then the column {v_1, v_2, ..., v_n} form a basis of R^n. d). If A and B are square matrices, then (A + B) (A - B) = A^2 - B^2. e). The rank of a lower-triangular matrix equals the number of non-zero entries in the diagonal. f). The rank of a lower-triangular matrix equals the number of non-zero entries in the diagonal. g). If A and B are square matrices such that AB invertible, then at least one of A or B is invertible but not necessarily both. h). If A = SDS^-1, then A^-100 = S^-1 D^-100 S. i). If A is invertible with A^10 = A^9, then A is the identity matrix. j). If A is a square matrix such that A^T A = I, then the columns of A are unit vectors.

Explanation / Answer

a)

Let us consider a homogeneous system of linear equations always has at least one solution that is the solution in which each unknown is zero. If the no of unknown values is greater than the equation then it has an infinite number of solution.

b) yes true but let us prove this by the reverse.

If rank (A) =n then A is an equivalent matrix to In×n. so there are some elementary matrices A1, A2,..., An such that Ans...A2A1A= I and so

A=A11...A1sI

But Ai1 are also invertible as I so A is invertible as well.

c) No

(AB)(A+B)=

A (A+B)B(A+B)=

A2+ABBAB2=

A2B2+ (ABBA).

But we know matrix multiplication is not commutative so ABBA.

So, (AB) (A+B)A^2-B^2

e) No

For upper triangular square matrix where all diagonal entries are zero, its rank will be bigger than zero, the number of non-zero diagonal elements.

Let us consider matrices example 2 by 2 and 3 by 3 matrices

f) No. We can simply consider that as above.

g) Yes true. We can say by matrix properties if AB is invertible then A and B are invertible but not necessarily both.

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