f undamentals of e ngineering a nalysis eng. hassan s. migdadi determinants. cramer’s rule part 5
TRANSCRIPT
FUNDAMENTALS OF ENGINEERING ANALYSIS
Eng. Hassan S. Migdadi
Determinants. Cramer’s RulePart 5
Properties of Determinants
1. What is the determinant of a triangular matrix?
2. How do elementary row operations effect the value of the determinant?
3. What is the determinant of an elementary matrix?
4. What is the determinant of an invertible matrix?
What is the determinant of a triangular matrix?
400
1620
927
Hint: Expand on column 1
Row Operations
Multiply a row by a non zero constant.
What happens to the determinant?
345
432
345
432
111 kkk
BA
Row Operations: Switch two rows
345
111
432
345
432
111
BA
Row Operations: Add a multiple of one row to another
345
432
111
345
432
111
kkkBA
Hint: Expand on Row 1
Theorem 1
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.
Example – Find |A|
2184
342
96 3
A
Strategy – Perform row operations to obtain an upper triangular matrix. Label each matrix with a new letter.
What is the determinant of an elementary matrix?
Suppose a matrix A is not invertible.
What can we say about det A?
Why?
Theorem 2: 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 two rows or columns are the same, then detA=0.
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.
Example :Find det A if
9085
6350
6350
5213
~
R32R1 R3
9085
4776
6350
5213
A
Theorem 3 – 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.
Theorem 4 – If A and B are nxn matrices, then
detAB = (detA)(detB)
Note - det(A+B)≠detA+detB