(3 times 9 marks) Mark each statement TRUE or FALSE. [] Every matrix is row equi

Question

(3 times 9 marks) Mark each statement TRUE or FALSE. [] Every matrix is row equivalent to a unique matrix in echelon form [] lf A is an m times n matrix and the equation Ax-b is consistent for every b in R", then A has m pivot columns. [] If A and B are row equivalent m times n matrices and if the columns of A span R", then so do the columns of B. [] If A and B are n times n matrices, then (A + B)(A - B) = A^2 - B^2. [] If A is a 3 times 3 matrix and the equation Ax = (1 0 0)^T has a unique solution, then A is invertible. [] If A^3 = 0, then det A = 0. [] If A is a 2 times 2 matrix with a zero determinant, then one column of A is a multiple of the other. [] If matrix A is invertible, then (adj A)^-1 =1/detA A. [] If A and B are square and invertible, then AB is invertible, and (AB)^-l = A^-1 B

Explanation / Answer

1) Every matrix is equivalent to matrix in echlon form --- True

2) If A is m xn matrix and Ax = b is consistent then reduced matrix A can have less than m pivots

and solution can be infinite --- False

4) (A+B)(A - B) = A^2 - AB + BA - B^2 which is not equal to A^2 - B^2

False

5) if Ax = b has a unique solution then A is invertible as x = A^-1b where b = [ 1, 0 , 0]^T

True

6) if A^3 =0 then det(A) =0 --- True

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