1. Determinants Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Here is the definition for 2x2 matrices:
The determinant of a square matrix can be calculated in a variety of ways.
It has many uses, one of which is to determine whether a matrix is invertible.
Here is the definition for 2x2 matrices:
For larger nxn matrices we can use a few different methods:
Cofactor Expansion
Row Reduction to Echelon Form
Basketweave Method (3x3 matrices only)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Cofactor Expansion Method
Here is a 3x3 example:
We will find the determinant of this matrix using all three methods.
Cofactor Expansion Method
This method breaks a larger square matrix into several smaller pieces, until eventually you have a bunch of 2x2 determinants to evaluate.
Choose a row or column to expand on.
Use a row or column with some zeroes for convenience.
Don’t forget to alternate signs, starting with + in the upper left.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Cofactor Expansion Method
Here is a 3x3 example:
We will find the determinant of this matrix using all three methods.
Cofactor Expansion Method
Here is what it looks like for this one, expanded on column one.
To find the sign of each term, use a checkerboard pattern:
Or use the formula (-1)(i+j)
add the row and column – if that is even, you get (+), if odd you get (-)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Row Reduction Method Row Operation Effect on Determinant
Here is a 3x3 example:
We will find the determinant of this matrix using all three methods.
Row Reduction Method
Using row operations, reduce the matrix to echelon form, then the determinant is the product of the diagonal elements. Keep track of the steps in the row reduction, and back out the effects to find the original determinant. Below is a table of the row operations and their effects.
Row Operation
Effect on Determinant
Add a multiple of a row to another row
No Change
Multiply a row by a constant k
Determinant is multiplied by k
Switch two rows
Determinant is multiplied by -1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Row Reduction Method Here is a 3x3 example:
We will find the determinant of this matrix using all three methods.
Row Reduction Method
Using row operations, reduce the matrix to echelon form, then the determinant is the product of the diagonal elements. Keep track of the steps in the row reduction, and back out the effects to find the original determinant.
In this example, the row reduction steps leave the determinant unchanged. We will do another example later.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Basketweave Shortcut for 3x3 determinants
Here is a 3x3 example:
We will find the determinant of this matrix using all three methods.
Basketweave Shortcut for 3x3 determinants
Recopy the first 2 columns next to the matrix, then multiply diagonals and add the blue, subtract the red as shown.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Here is another example:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. Cofactor Expansion Method
Here is another example:
Cofactor Expansion Method
I will choose row 4 to expand, since it has 2 zeroes. Could also use column 3.
For each of these 3x3 determinants I will use the basketweave method.
Adding it all up we get det(B) = -4(6)-1(-2)=-22
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. Row Reduction Method Here is another example:
The determinant is the product of the diagonal elements = (1)(4)(-1)(11)=-44
We have to undo the effects of our row operations. The only step that changed the value of the determinant was 2R3-3R2. This is really two steps in one: multiply row 3 by 2, then subtract 3 row 2’s from that.
It’s the multiplying by 2 that we have to undo
So det(B) = (-44)/2 = -22.
Same answer we got using the other method.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. Here are a few convenient rules for determinants:
det(AB) = det(A)det(B)
det(AT)=det(A)
det(A-1) = 1/det(A)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB