1. Determinant of a Matrix

2. Determinant of a Matrix

3. Determinant of a Matrix

4. Properties of Determinants
The determinant of a matrix and its transpose are equal, that is |AT| = |A|.
If a matrix B results from a matrix A by interchanging two rows (columns) of A, then |B| = −|A|.

5. Properties of Determinants
If two rows (columns) of a matrix A are equal, then |A| = 0.
If a row (column) of a matrix A consists entirely of zeros, then |A| = 0.

6. Properties of Determinants
If a matrix B is obtained from a matrix A by multiplying one row (column) of A by a number r , then |B| = r |A|.
If a matrix B is obtained from a matrix A by adding a multiple of one row (column) of A to another row (column) of A, then |B| = |A|.

7. Properties of Determinants
The determinant of an n×n upper (lower) triangular matrix is the product of its main diagonal entries.
The determinant of an n×n diagonal matrix is the product of its main diagonal entries.
|AB| = |A| |B |.
|An| = |A|n
If A is an n×n matrix and r is a scalar , then |r A| = r n |A|.
A is nonsingular if and only if |A| ≠ 0.

8. Problems
Example 1: Evaluate

9. Problems
Example 2: If A and B are 5×5 matrices with |A| = 8 and |B| = 2. Find
(i) |A2| (ii) |-A| (iii) |AT B -1| (iv) |2A-1B 4A|.

10. Problems
Example 3: Answer each of the following as True or False. Justify your answer.
(i) If A and B are n×n matrices, then |AB | = |BA|.
(ii) If A and B are n×n matrices, then |A + B | = |A| + |B |.
(iii) If A and B are n×n matrices such that AB2 = In , then A is nonsingular.

11. Problems
Example 4: By reduction to triangular form, evaluate

12. Problems
Example 5: