wl02727399
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matrix determinant
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Do you know the "properties of a determinant of a matrix". Find those in your text-book and make list - you will use those.
The determinant of a matrix "does not change" if a interchange two rows of a matrix.
The determinant of a matrix "does not change" if a interchange two columns of a matrix.
Start with the matrix (whose determinant you need to calculate) and interchange the first and third rows (idea here is to make it look like the matrix whose determinant you already know).
Continue.....
Please show us what you have tried and exactly where you are stuck.
Please follow the rules of posting in this forum, as enunciated at:
https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520
Please share your work/thoughts about this assignment.
HallsofIvy
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My first thought was "tedious, yes, Calculating determinants always is." But, if you know what the "properties of determinants" are this is really very easy. You don't have to actually evaluate any 3 by 3 determinant!
We are given that \(\displaystyle \left|\begin{array}{ccc}5 & x & 7 \\ 10 & y & 3 \\ 20 & 17 & 7 \end{array}\right|= -385\). The first problem asks you to find the determinant \(\displaystyle \left|\begin{array}{ccc} 4 & 17 & 7 \\ 2 & y & 3 \\ 1 & x & 7\end{array}\right|\).
Do you see that the third column of this matrix is the same as the third column of the first? But since it is "symmetric" it is also the third column reversed? And that the second column is just the second column of the first reversed? And that the first column is one fifth the first column of the previous matrix, also reversed? So you can find the determinant of the second matrix without actually expanding it!
Of course, that will be an equation in x and y. And setting y= 12, you have an equation to solve for x.
We are given that \(\displaystyle \left|\begin{array}{ccc}5 & x & 7 \\ 10 & y & 3 \\ 20 & 17 & 7 \end{array}\right|= -385\). The first problem asks you to find the determinant \(\displaystyle \left|\begin{array}{ccc} 4 & 17 & 7 \\ 2 & y & 3 \\ 1 & x & 7\end{array}\right|\).
Do you see that the third column of this matrix is the same as the third column of the first? But since it is "symmetric" it is also the third column reversed? And that the second column is just the second column of the first reversed? And that the first column is one fifth the first column of the previous matrix, also reversed? So you can find the determinant of the second matrix without actually expanding it!
Of course, that will be an equation in x and y. And setting y= 12, you have an equation to solve for x.