Answer to Question #269890 in Linear Algebra for john

Question #269890

Question 4 Use row operation to show that

det T = 0


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x

2 2x + 1 4x + 4 6x + 9

y

2 2y + 1 4y + 4 6y + 9

z

2 2z + 1 4z + 4 6z + 9

w

2 2w + 1 4w + 4 6w + 9

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1
Expert's answer
2021-11-24T10:08:43-0500

"|T|=\\begin{vmatrix}\n x^2 & 2x+1 & 4x+4 &6x + 9\\\\\n y^2 & 2y+1 & 4y+4 &6y + 9\\\\\nz^2 & 2z+1 & 4z+4 &6z + 9\\\\\nw^2 & 2w+1 & 4w + 4&6w + 9\n\\end{vmatrix}"


if we add 2nd and 3rd columns, and subtract it from 4th column, we get:


"|T|=\\begin{vmatrix}\n x^2 & 2x+1 & 4x+4 & 4\\\\\n y^2 & 2y+1 & 4y+4 &4\\\\\nz^2 & 2z+1 & 4z+4 &4\\\\\nw^2 & 2w+1 & 4w + 4&4\n\\end{vmatrix}"


if we multiply 2nd column by 2, and subtract it from 3rd column, we get:


"|T|=\\begin{vmatrix}\n x^2 & 2x+1 & 2 & 4\\\\\n y^2 & 2y+1 & 2 &4\\\\\nz^2 & 2z+1 & 2 &4\\\\\nw^2 & 2w+1 & 2&4\n\\end{vmatrix}"


then, multiplying 3rd column by 2, we get:


"2|T|=\\begin{vmatrix}\n x^2 & 2x+1 & 4 & 4\\\\\n y^2 & 2y+1 & 4 &4\\\\\nz^2 & 2z+1 & 4 &4\\\\\nw^2 & 2w+1 & 4&4\n\\end{vmatrix}=0"


since two columns of a determinant are identical.


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