Let A,B ∈ Matn(F ) denote two square matrices. Assume that A and B are row equivalent. Show that B is invertible if and only if A is invertible.

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Please show all work for the proof

Assume that T is a linear transformation. T: Projects vectors in its domain onto the line y = 4x. Find the standard matrix of T. Enter your matrix as 4-tuple (a11, 12, 021, 922) Use only fractions in simplest form and integers. NO DECIMALS "a11 11 "),a12= ,a21= ,a22=

Let A and P be a nxn matrices and P is invertible. Show that A and PA have the same rank

Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are nonzero.

Please show how to determine whether matrix A is diagonalizable: If possible, find an invertible P and a diagonal D such that P-1AP=D

Let A and B be m x n matrices. (a) Suppose that v"Aw A = B. v" Bw for all vectors v, w with real entries. Prove that