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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 determinants

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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,

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we multiply the entries
in the main diagonal

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and we subtract from that
the product of the entries

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in the off diagonal.

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

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minus, minus 4 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

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going to use a Laplace
expansion, which

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means that we're going to need
to choose a row or a column

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in the matrix.

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

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I'd like to choose
the first row,

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because I see this 0 here,
which is going to mean

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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

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by 2 determinant, which
we get by covering up

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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

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is cover up the row and
column containing that,

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and we have 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--
just for practice,

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why don't I put in this
cofactor here anyways.

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So again, we cover up
the row and the column

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containing the 0, and we
have this matrix 1, 2; 3, 1.

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Now finally, we have
to walk over here,

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and we have to take 4
times the minor-- which

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we get by covering up the row
and column containing 4-- 1,

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2; 3, minus 2.

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And now notice that these
are just 2 by 2 determinants

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

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Altogether, we get
minus 1, times 2

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minus another 2-- excuse me, 2
minus-- 2 minus a negative 4,

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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

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have minus 2 minus 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 on the 3

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by 3 matrix.

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We needed to do a
Laplace expansion, which

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means that we needed to
choose a row or a column.

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And we needed to take
the entries of the row

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and add these up, multiplied
by the cofactor matrix

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we got by 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,

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minus 0 times this cofactor,
and then finally, plus 4 times

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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.