Mark each statement True or False. (You don\'t need to justify the reason): a:__

Question

Mark each statement True or False. (You don't need to justify the reason): a:__ If an m times n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in R^m. b:__ If A is a 6 times 5 matrix, the linear transformation x rightarrow Ax. cannot map R^6 onto R^5. c:__ If A is 4 times 4 matrix, then det (4A) = 4 det A d:__ If AB = AC. then B = C, where A, B and C are matrices with suitable dimensions (suitable dimensions mean that the products are well defined). e:__ If A and B are n times n matrices, with det A = 2 and detB = 3, then det(A + B) = 5. f:__ If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.

Explanation / Answer

a) True

Since, there is a pivot point in each row, Hence, there is unique solution for each b in R^m.

b) True

The system Ax = B with a 6 x 5 co-efficient matrix A has atmost 5 pivot points. Hence, cannot be a solution for each b belongs to R^6.

c) False

If A is a nxn matrix, then

Det(mA) = m^n det(A)

d) True

AB = AC

=> A-1 AB = A-1 AC

=> IB = IC

e) det (A + B) = det(A) + det(B)

=> B = C

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