1
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As another hint, I will take the same matrix, matrix A

2
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and take its determinant again but I will do it using a different technique, either technique is valid

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so here we saying what is the determinant of

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the 3X3 Matrix A and we can is we can rewrite first two column

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so first column right over here we could rewrite it as 4 4 -2 and then the second column right over here

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we could rewrite it

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-1 5 0

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and we could do is we could take the sum of the products

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of the first three top left bottom left diagonal,

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let me show you

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so the product of that

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that

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plus

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that

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plus

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that

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trying my best to draw this neatly

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and then from that subtract

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top right to bottom left diagonal

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so from that subtract

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let get a color

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I havent used

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subtract that

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and that

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and that

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sounds really confusing with all the diagonals

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i have drawn

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lets look at the blue ones first

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4 times 5 times 0

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4 times 5 times 0

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plus -1 times 3 times -2

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plus -1 times 3 times -2

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let me put these in parenthesis

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plus 1 times 4 times 0

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and then we gonna subtract all of these orange

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diagonal, we go from the top right to the bottom left

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so we gonna subtract

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we could do 1 times 5 times -2

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1 times 5 times -2 and then we can subtract

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subtract 4 times 3 times 0

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4 times 3 times 0

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and then we can subtract

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-1 times 4 times 0

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-1 times 4 times 0

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and now we just evaluate this over here

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4 times 5 times 0

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is just 0

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-1 times 3 times -2

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is +6 so this is +6

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1 times 4 times 0 is 0 once again

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and then we have

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1 times 5 times -2

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is -10 but we have this negative out here so it becomes a +10

54
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then 4 times 3 times , well thats just going to be 0.

55
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and then we have -1 times 4 times 0, which is just 0

56
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so we are left with +6 + 10

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which is equal to +16