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Properties of Determinants

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**1. **Properties of Determinants What is the determinant of a triangular matrix?
How do elementary row operations effect the value of the determinant?
What is the determinant of an elementary matrix?
What is the determinant of an invertible matrix?

**2. **What is the determinant of a triangular matrix?

**3. **Row Operations Multiply a row by a non zero constant.
What happens to the determinant?

**4. **Row Operations: Switch two rows

**5. **Row Operations: Add a multiple of one row to another

**6. **Theorem 3 Multiplication of a row by a constant multiplies the determinant by that constant.
Switching two rows changes the sign of the determinant.
Replacing one row by that row plus a multiple of another row has no effect on the determinant.
Proof by induction is given in the text.

**7. **Example – Find |A|

**9. **Suppose a matrix A is not invertible.
What can we say about det A?
Why?

**10. **Theorem 4: A is invertible iff detA?0.
Note – This theorem links the determinant to the invertible matrix theorem.
For instance, if the columns (or rows) of A are linearly dependent, then detA=0.
So if you perform row operations so that 2 rows or columns are the same, then detA=0.

**11. **Proof (outline) A is invertible iff A is row equivalent to In.
iff detA?0
Note that each row operation changes the determinant by some non zero factor.
Since det In=1, we couldn’t have started with a determinant of 0.

**12. **Example 3 (from text) Find det A if

**13. **Theorem 5 – If A is an nxn matrix,
detAT=detA
Proof: By induction. Theorem is obvious for n=1.
Suppose it is true for n=k. Let n=k+1.
The cofactor of a1j in A equals the cofactor of aj1 in AT because the cofactors involve kxk determinants and we’ve assumed the theorem is true for n=k.
So the cofactor expansion along the first row of A equals the cofactor expansion along the first column of AT.
By the principle of induction, the theorem is true for all n=1.

**14. **Theorem 6 – If A and B are nxn matrices, then
detAB = (detA)(detB)
Proof – Please read in text for homework.
Note - det(A+B)?detA+detB

**15. **Homework – Finish reading and exercises for sections 3.1 and 3.2.