Linear Algebra True or FalseSolutionPlease post one more que

Linear Algebra: True or False

Solution

Please post one more question to get the answers to remaining parts, i have provided answers to three parts in detail. Thanks,Nikhil

c) If AB = AC and A is invertible, then B=C

(Statement is TRUE)

Since given A is invertible hence A^(-1) exists multiplying both sides by A^(-1) we get

A^(-1)[AB] = A^(-1)[AC]

[A^(-1)A]B = [A^(-1)A]C => IB = IC ===> B=C

d) The statement is correct according to the given below proof

Since the matrix inverse A exists being invertible, let the inverse be A^(-1)

Since k is not equal to zero as per question hence 1/k exists

[1/k*A^(-1)] * [KA] => A^(-1)A = I

Hence the given KA is invertible and has inverse equal to [1/k*A^(-1)]

e) Every elemental matrix is invertible ---- The given statement is TRUE

Since assume there exists an operation O1, which will change the elemental matrix to the required order (nXn) Identity matrix, If we perform the operation O1 on the identity then we will get the inverse of the matrix.

Since inverse exists, hence elemental matrix is invertible