co. chapter 3 determinants linear algebra. ch03_2 let a be an n n matrix and c be a nonzero scalar....
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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