Linear Algebra
Co. Chapter 3
Determinants
3.2 Properties of Determinants
Theorem 3.2
Let A be an n  n matrix and c be a nonzero scalar.
(a) If A  B then |B| = ……..
cR i
(b) If A
(c) If A
Ri
 B then |B| = ….....
R
j
 B then |B| = …….
R i cR j
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Example 1
4 3
 1
2 5, |A| = 12 is known.
If A   0
 2  4 10
Evaluate the determinants of the following matrices.
4 3
 1 12 3
 1
 1 4 3
(a ) B1   0
6 5 (b) B2   2  4 10 (c) B3  0 2 5
 2  12 10
 0
0 4 16
2 5
Solution
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Definition
A square matrix A is said to be …………. if |A|=0.
A is …………….. if |A|0.
Theorem 3.3
Let 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 …………….. (……………..)
Example 3 : Show that the following matrices are singular.
 2 0  7
2  1 3
(a ) A   3 0
1 (b) B   1 2 4
 4 0
2 4 8
9
Solution
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Theorem 3.4
Let A and B be n  n matrices and c be a nonzero scalar.
(a) |cA| =………
(b) |AB| =………
(c) |At| =……….
1
A
 ...........
(d)
(assuming A–1 exists)
Ch03_5
Example 4
If A 22 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
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Example 6
Prove 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 ......................
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3.3 Numerical Evaluation of a Determinant
Definition
A square matrix is called an upper (lower) triangular matrix if
all the elements below (above) the main diagonal are zero.
3
0

0
1 4 0
8 2 
0 2 3


1 5 ,
0 0 0
0 9  
0 0 0
............  triangular
7
5 
9

1
8 0 0
7 0 0  
 2 1 0  , 1 4 0

 7 0 2
 3 9 8 
4 5 8
.............  triangular
0
0 
0

1
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Numerical Evaluation of a Determinant
Theorem 3.5
The determinant of a triangular matrix is the ………… of its
main diagonal elements.
Example 1
2 1 9
Let A  0 3 4 , find A .
0 0 5
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Numerical Evaluation of a Determinant
Example 2
Evaluation the determinant.
Solution
Example 3
Evaluation the determinant.
Solution
 2 4 1
A   2 5 4 
 4 9 10 
 1 2 4 
B   1 2 5
 2 2 11
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Example 4
1 1 0 3
Evaluation the determinant.
1
1 2 3
2 2 3 4
Solution
6 6 5 1
Ch03_11