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