Co. Chapter 3

Determinants

Linear Algebra

Ch03_2

Let A be an n n matrix and c be a nonzero scalar.

(a) If then |B| = ……..

(b) If then |B| = ….....

(c) If then |B| = …….

3.2 Properties of DeterminantsTheorem 3.2

icRA B

i jR RA B

i jR cRA B

Ch03_3

Example 1

If |A| = 12 is known.

Evaluate the determinants of the following matrices.

,1042

520341

A

1640

520

341

)c(

520

1042

341

(b)

10122

560

3121

)a( 321 BBB

Solution

Ch03_4

Theorem 3.3Let A be a square matrix. A is singular if

(a) All the elements of a row (column) are …………

(b) two rows (columns) are ……………..

(c) two rows (columns) are …………….. (……………..)

DefinitionA square matrix A is said to be …………. if |A|=0.

A is …………….. if |A|0.

Example 3 : Show that the following matrices are singular.

842

421

312

(b)

904

103

702

)(a BA

Solution

Ch03_5

Theorem 3.4Let A and B be n n matrices and c be a nonzero scalar.

(a) |cA| =………

(b) |AB| =………

(c) |At| =……….

(d) (assuming A–1 exists) 1 ...........A

Ch03_6

Example 4If matrix with |A| = 4, compute the following determinants.(a) |3A| (b) |A2| (c) |5AtA–1|, assuming A–1 exists

Solution

(a) |3A| = …………………………..……..

(b) |A2| = ………………………………….

(c) |5AtA–1| = ……………………………..

Example 5

Prove that |A–1AtA| = |A|

Solution

2 2A

Ch03_7

Example 6Prove that if A and B are square matrices of the same size, with A being singular, then AB is also singular. Is the converse true?

Solution

Note:

......................A B

Ch03_8

3.3 Numerical Evaluation of a Determinant

DefinitionA square matrix is called an upper (lower) triangular matrix if all the elements below (above) the main diagonal are zero.

1 4 0 73 8 2

0 2 3 50 1 5 ,

0 0 0 90 0 9

0 0 0 1

............ triangular

8 0 0 07 0 0

1 4 0 02 1 0 ,

7 0 2 03 9 8

4 5 8 1

............. triangular

Ch03_9

2 1 9

Let 0 3 4 , find .

0 0 5

A A

Theorem 3.5

The determinant of a triangular matrix is the ………… of its main diagonal elements.

Example 1

Numerical Evaluation of a Determinant

Ch03_10

Numerical Evaluation of a DeterminantExample 2

Evaluation the determinant.

2 4 1

2 5 4

4 9 10

A

Evaluation the determinant.

2 4 1

2 5 4

4 9 10

A

Solution

Example 3

Evaluation the determinant.

1 2 4

1 2 5

2 2 11

B

Solution

Ch03_11

Example 4

Evaluation the determinant.

1 1 0 3

1 1 2 3

2 2 3 4

6 6 5 1

Solution