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Determinants

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- All square matrices have a determinant. The only ones we will deal with are 2x2 and 3x3 matrices.
- To find the determinant of a 2x2 matrix, you use the rule for second order determinants.
- ad-bc
- There are two ways to find the determinant of a 3x3 matrix.
- First is expansion by minors. To get your minors, pick any row of the matrix look at the first digit, it will become a multiple for a 2x2 matrix. Ignore the other numbers in the row and cover up the other numbers in the column. You will only have 4 numbers left in the matrix and take that as a 2x2 matrix. Use the digit you started with as a multiple of the matrix. Repeat those steps for the remaining two digits in the row. Look at the next page for the formula.

- This is assuming you pick the first row for your multiples but remember that you can pick any row.
= a* -b* +c*

- Observe how when a is the multiple, the corresponding matrix is only the elements left over when the row and column of a are covered.
- The other way to find the determinant of a 3x3 matrix is by using diagonals. Look at the next page to see how it works.

First take the first two columns and add them on to the end.

Then draw diagonals from the first entry of each row down and to the right. You obtain aef, bfg, and cdh. Then start at the bottom and draw diagonals up and to the right. You get gec, hfa, and idb.

The determinant will equal aef+bfg+cdh-gec-hfa-idb.

Solve the following:

1)2)3)

Solve by expansion by minors:

4)5)6)

Solve by diagonals:

7)8)9)

- Answers:
- 1) 142)-223)-224)-325)-956) 3697)-3238)-309) 90