In each part determine whether the homogeneous system has anon trivial solution

Question

In each part determine whether the homogeneous system has anon trivial solution (without using pencil and paper, explain);then state whether the given matrix is invertible. a) 2x1 + x2 - 3x3 +x4 = 0     [ 2 1 -3 1]             5x2+ 4x3 + 3x4=0   [0   5   4 3 ]                       x3 + 2x4 = 0   [ 0 0   1 2]                               3x4= 0   [ 0 0   0 3] I know this only has non trivial solutions and is invertible.I don't know how to tell just by looking at the matrix if someonecould explain to me how I can tell just by looking at a matrixif it is invertible or not invertible. Also if the matrix has onlytrivial solutions or infinitly many, just by looking at the givenmatrix. thankyou in advance. In each part determine whether the homogeneous system has anon trivial solution (without using pencil and paper, explain);then state whether the given matrix is invertible. a) 2x1 + x2 - 3x3 +x4 = 0     [ 2 1 -3 1]             5x2+ 4x3 + 3x4=0   [0   5   4 3 ]                       x3 + 2x4 = 0   [ 0 0   1 2]                               3x4= 0   [ 0 0   0 3] I know this only has non trivial solutions and is invertible.I don't know how to tell just by looking at the matrix if someonecould explain to me how I can tell just by looking at a matrixif it is invertible or not invertible. Also if the matrix has onlytrivial solutions or infinitly many, just by looking at the givenmatrix. thankyou in advance.

Explanation / Answer

QuestionDetails: In each part determine whether the homogeneous system has anon trivial solution
(without using pencil and paper, explain); then state whetherthe given matrix is invertible. a) 2x1 + x2 - 3x3 +x4 = 0     [ 2 1 -3 1]             5x2+ 4x3 + 3x4=0   [0   5   4 3 ]                       x3 + 2x4 = 0   [ 0 0   1 2]                               3x4= 0   [ 0 0   0 3] I know this only has non trivial solutions[NOT CORRECT]and isinvertible[CORRECT]. I don't know how totell just by
looking at the matrix if someone could explain to me howI can tell just by looking at a matrix
if it is invertible or not invertible. Also if the matrix has onlytrivial solutions or infinitly many,
just by looking at the given matrix.
YOU ARE CORRECT IN THAT , IN GENERAL WE MAY NOT BE ABLE TO SAY
THAT BY A LOOK AT THE MATRIX.
BUT HERE THE MATRIX GIVEN IS OF SPECIAL TYPE.IT IS UPPERTRIANGULAR.
SO WE CAN TELL BY LOOKING AT IT
IF THERE IS NO ROW WITH ALL ZEROS IN THIS UPPER TRIANGULAR FORM
THEN THE DETERMINANT OF THE MATRIX IS NON ZERO AND HENCE THE
MATRIX IS INVERTIBLE.
AS REGARDS NON TRIVIAL/TRIVIAL SOLUTIONS,LOOK AT THE CONSTANTS.
THEY ARE ALL ZEROS . SO IF THE DETERMINANT OF THE MATRIX IS NONZERO
AND THE MATRIX IS INVERTIBLE , AS IS THE CASE HERE,WE GET ANUNIQUE
SOLUTION WHICH IS OBVIOUSLY THE TRIVIAL SOLUTION.
THAT IS X1=X2=X3=X4=0
IF THE DETERMINANT OF THE MATRIX IS ZERO AND THE MATRIX ISNOT INVERTIBLE ,
WE GET INFINITE SOLUTIONS.

thankyou in advance.

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