A 3 times 3 matrix A contains two equal rows State whether e

A 3 times 3 matrix A contains two equal rows. State whether each of the following is true or false. (a) A has an inverse. (b) The rows of A are linearly independent vectors. (c) The determinant of A is equal to zero. (d) The equation Ax. = b, where x = (x_1, x_2, x_3)^T and b = (1, -1, 1)^T, has no solutions for x_1,x_2 and x_3. (e) The rows of A form a basis for R^3 (f) The rank of A is 3.

Solution

(a) False as det (A) = 0

(b) False as one row is a linear combination of the other row , with coefficients 1 , 1 , 1.

(c) True

(d) False, as it has not been stated that the equal rows are two consecutive rows. If the equal rows are two conseqcutive rows, then the statement is true. For example if the equal rows ( a , b, c) are consecutive, then the equations ax₁ + bx₂ + cx₃ = 1 and ax₁ + bx ₂+ cx₃ = -1 are inconsistent., otherwise, the equations ax₁ + bx₂ + cx₃ = 1 and dx₁ + ex ₂+ fx₃ = -1 ( being two equations in 3 variables) can be consistent and can have infinite solutions.

(e) False. Since the rows of A are not linearly independent, these will not form a basis for R^3.

( f ) False.. There being a maximum of 2 linearly independent rows, the RREF of A will necessarily have a zero row. Therefore, the maximum possible rank of A is 2.