1. Let A be an n × n matrix and b be an n × 1 matrix, which of the following sta
ID: 3283829 • Letter: 1
Question
1. Let A be an n × n matrix and b be an n × 1 matrix, which of the following statements is/are true? (i) Ax = b has a unique solution if and only if b can be expressed as a linear combination of the columns of the matrix A. (ii) A is singular if and only if A can be expressed as a product of elementary matrices. (ii) Ax 0 has only the trivial solution if and only if the reduced row echelon form of A is In (iv) Ax b has a unique solution if and only if det(A) A. (i) only B. (i) only C. (i) and (ii) only D. (i) and (iv) only E. (i), (ii) and (ii) only 2. Let A, B be an n x n invertible matrix, which of the following statements is/are always true? (ii) (A"B")-1-(B-1)7(A-1)3 (ili) det(kA)-kdet (A) for any scalar k (iv) (kA)-1-k-A-1 for any scalar k0 (v) A +AT is diagonalizable A. (i) and (ii) only B. (i) and (iv) only C. (ii), (iv) and (v) only D. (ii), (iv) and (v) onlyExplanation / Answer
Answer for Question 1. For other question please post it separately.
1
(i)
Let's consider the first statement:
Ax = b is consistent if and only if b is the linear combination of the column vectors of A. Being consistent doesn't imply unique solution. The condition is necessary but not sufficient. The statement is true.
(ii)
If Ax = b is an n × n matrix then A is non-singular if and only if A is the product of elementary matrices. Hence, (ii) is false.
(iii)
Let A be nxn matrix . Then, the system AX = b has a ( unique ) solution if its reduced row echelon form is an identity matrix of order n.
Hence, (iii) is true.
(iv)
To have a unique solutions the deteminant A must not be zero. Hence, (iv) is false.
Hence, the ansewer is C: (i) and (iii).
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