1. F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Determinants. Cramer’s Rule Part 5

2. 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?

3. What is the determinant of a triangular matrix? Hint: Expand on column 1

4. Row Operations Multiply a row by a non zero constant. What happens to the determinant?

5. Row Operations: Switch two rows

6. Row Operations: Add a multiple of one row to another Hint: Expand on Row 1

7. 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.

8. Example – Find |A| Strategy – Perform row operations to obtain an upper triangular matrix. Label each matrix with a new letter.

9. What is the determinant of an elementary matrix?

10. Suppose a matrix A is not invertible. What can we say about det A? Why?

11. 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.

12. Proof (outline) A is invertible iff A is row equivalent to I n. iff detA≠0 Note that each row operation changes the determinant by some non zero factor. Since det I n =1, we couldn’t have started with a determinant of 0.

13. Example :Find det A if

14. Theorem 3 – If A is an nxn matrix, detA T =detA Proof: By induction. Theorem is obvious for n=1. Suppose it is true for n=k. Let n=k+1. The cofactor of a 1j in A equals the cofactor of a j1 in A T 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 A T. By the principle of induction, the theorem is true for all n≥1.

15. Theorem 4 – If A and B are nxn matrices, then detAB = (detA)(detB) Note - det(A+B)≠detA+detB