Q1. In each part of the following determine whether the statement is true or fal
ID: 3115715 • Letter: Q
Question
Q1. In each part of the following determine whether the statement is true or false. Justify your answer. 1- If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions. 2- If A and B are two square matrices of the same order, then tr(2A+ AB*) = 2tr(A) + tr(A)tr(B) 3- If B has a column of zeros, then so does AB if this product is defined. 4- The sum of two invertible matrices of the same size is invertible. 1 3 0 01 5-The matrix A is in reduced row echelon form. L0 0 0 0 6- If A and B are invertible matrices of the same size that satisfies 7- If A and B are square matrices of the same size and k is a constant, then 8- If A is an n × n matrix that is not invertible, then the matrix obtained by 9- Let A and B be n × n matrices. If AB is invertible, then both A and B must A(A1 B1)B(A + B)-1I, then (A B)-1-B+ A. (3A + 5B)2 = 9A2 + 30AB + 25B2. interchanging two rows of A cannot be invertible. be invertible.Explanation / Answer
(1) Since the rank of the marix is directly equal to the no of non zero rows of the matrix reduced in to echelon form. therefore no conclusion can be made by this formany solution. hence the statement is false.
(2) The trace of matrix is equal to the sum of elements lying along the principal diagonal. therefore the statement is false because trace of matrix(2A +ABT = 2 TR(A) + TR(A) tr(B).
(3) Since B has a zero column and AB also has a zero column. therefore no of columns in matrix is not equal to the no of rows of the matrix B therefore the product is not defined.
(4) Thesum of two invertible matrices is always invertible. true according the theorem of matrices.
(5) Since in row echelon form if all the element lying below the principal diagonal is zero and leading element of every row should be one. according this this the given statement is false
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