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DAVID JORDAN: Hello, and welcome
back to recitation.

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As a warm up, let's get started
by computing some

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determinants for 2 by 2
and 3 by 3 matrices.

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Why don't you take some time to
work on computing these two

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determinants, and when you're
finished, check back with me

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and I'll show you
how I solved it.

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Welcome back.

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So why don't we get started with
the 2 by 2 matrix first.

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So remember, when we compute
a 2 by 2 determinant, we

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multiply the entries in the main
diagonal and we subtract

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from that the product of the
entries in the off diagonal.

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So in this case, we have 3 times
minus 2, minus, minus 4

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times minus 1.

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So we have minus 6 minus
4 is minus 10.

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OK, now, the 3 by 3 matrix,
we're going to use a Laplace

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expansion, which means that
we're going to need to choose

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a row or a column
in the matrix.

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We can choose any row or column,
but as I look at this

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matrix, I'd like to choose the
first row, because I see this

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0 here, which is going to mean
we have less work to do.

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So let's do Laplace expansion
across the first row.

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So what that means is we take
the very first entry, minus 1,

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and now we need to multiply it
by a 2 by 2 determinant, which

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we get by covering up the row
and the column corresponding

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to our first entry.

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So our first entry was minus 1,
and what we need to do is

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cover up the row and column
containing that, and we have

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this little 2 by
2 matrix here.

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And so we get 2,
2, minus 2, 1.

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OK.

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The next entry, we have to take
negative of this entry,

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but this entry is 0.

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So minus 0 times--

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just for practice, why
don't I put in this

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cofactor here anyways.

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So again, we cover up the row
and the column containing the

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0, and we have this
matrix 1, 2, 3, 1.

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Now finally, we have to walk
over here, and we have to take

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4 times the minor--

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which we get by covering
up the row and

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column containing 4--

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1, 2, 3, minus 2.

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And now notice that these are
just 2 by 2 determinants and

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we can just compute those the
same way we did earlier.

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Altogether, we get minus 1,
times 2 minus another 2--

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excuse me--

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2 minus, 2 minus a negative
4, so we get 6.

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OK, this one goes away.

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And then we have plus 4, times,
we have minus 2 minus

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another 6, so it looks
to me like minus 8.

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Altogether, we have minus 38.

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Now let's just take a moment
to see what we did

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on the 3 by 3 matrix.

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We needed to do a Laplace
expansion, which means that we

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needed to choose a
row or a column.

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And we needed to take the
entries of the row and add

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these up, multiplied by the
cofactor matrix we got by

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covering up the row and
column containing

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our highlighted entry.

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And we needed to do that
alternating the signs.

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So we got minus 1 times this
cofactor, minus 0 times this

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cofactor, and then finally, plus
4 times this cofactor.

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Altogether, we got minus 38.

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OK, I'll leave it at that.

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Thank you.

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