Determinants matriks 3x3 cara kofaktor
26
Determinant of a Matrix using Co-factors and Minors By: Jeffrey Bivin Lake Zurich High School [email protected] Last Updated: October 11, 2005
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Transcript of Determinants matriks 3x3 cara kofaktor
Determinant of a Matrix using
Co-factors and Minors
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 11, 2005
Determinant of a 3 x 3 Matrix
Choose arow or
a colum
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
We will use the first column to give us our
cofactors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Notice the alternating signs
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Now for the minors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
Remove the row and the column of the cofactor element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
Remove the row and the column of the cofactor element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Remove the row and the column of the cofactor element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 51(7 – 30) – 2(21 – 24) + 3(15 – 4)
1(-23) – 2(–3) + 3(11) = -23 + 6 + 33 = 16
= 16Evaluate each 2x2 determinant and
simplify
Jeff Bivin -- LZHS
Now try a different row or column
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
Choose a newrow or
a colum
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
This time we will use the second row to give
us our cofactors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Again we have alternating signs
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Now for the minors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the column of the cofactor element
3 4
6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the column of the cofactor element
3 4
6 7
1 4
3 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the column of the cofactor element
3 4
6 7
1 4
3 7
1 3
3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
3 4
6 7
1 4
3 7
1 3
3 6-2(21 – 24) + 1(7 – 12) - 5(6 – 9)
-2(-3) + 1(–5) - 5(-3) = 6 - 5 + 15 = 16
= 16Evaluate each 2x2 determinant and
simplify
Jeff Bivin -- LZHS
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
Select a row or column to use as the co-factors.
Jeff Bivin -- LZHS
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
Let’s use the first row for the co-factors
Jeff Bivin -- LZHS
1 3 4-2
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
= 16Remember the
alternating signs.
Jeff Bivin -- LZHS
1 -2 3 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
+ 3 - 4
Remove the row and the column of the cofactor element
1 + 2
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
1 + 22 1 5
2 -1 4
3 6 7
Remove the row and the column of the cofactor element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
1 + 22 -5 5
2 3 4
3 1 7
2 1 5
2 -1 4
3 6 7
Remove the row and the column of the cofactor element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
2 1 5
2 -1 4
3 6 7
2 -5 5
2 3 4
3 1 7
2 -5 1
2 3 -1
3 1 6
1 + 2
Remove the row and the column of the cofactor element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
2 1 5
2 -1 4
3 6 7
2 -5 5
2 3 4
3 1 7
2 -5 1
2 3 -1
3 1 6
1 + 2
Now evaluate the 3x3 determinants --- more expansion by
co-factors and minors
1(233) + 2(11) + 3(9) - 4(106) = -142
+ 3 - 4
= -142
