Discrete Mathematics - Matrices & Determinants

Matrices

• Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices.

• Operation on matrices: Addition and multiplication and multiplication with a scalar

• Simple properties of addition, multiplication and scalar multiplication

• Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2)

• Concept of elementary row and column operations

• Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

Determinants

• Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle

• Ad joint and inverse of a square matrix

• Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix

SUMMARY

Matrices

1. A matrix is an ordered rectangular array of numbers or functions.

2. A matrix having m rows and n columns is called a matrix of order m × n.

3. An m × n matrix is a square matrix if m = n.

4. A = [aij] = [bij] = B if (i) A and B are of same order, (ii) aij = bij for all possible values of i and j.

5. – A = (–1)A

6. A – B = A + (–1) B

7. A + B = B + A

8. (A + B) + C = A + (B + C), where A, B and C are of same order.

9. k(A + B) = kA + kB, where A and B are of same order, k is constant.

10. (k + l) A = kA + lA, where k and l are constant.

11. (i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC

12. A is a symmetric matrix if A′ = A.

13. A is a skew symmetric matrix if A′ = –A.

14. Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

15. If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and A is the inverse of B.

16. Inverse of a square matrix, if it exists, is unique.

Determinants

1. For any square matrix A, the |A| satisfy following properties.

2. |A′| = |A|, where A′ = transpose of A.

3. If we interchange any two rows (or columns), then sign of determinant changes.

4. If any two rows or any two columns are identical or proportional, then value of determinant is zero.

5. If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.

6. Multiplying a determinant by k means multiply elements of only one row (or one column) by k.

7. If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.

8. If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.

9. Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting i th row and j th column and denoted by Mij.

10. A (adj A) = (adj A) A = |A| I, where A is square matrix of order n.

11. A square matrix A is said to be singular or non-singular according as |A| = 0 or |A| ≠ 0.

12. If AB = BA = I, where B is square matrix, then B is called inverse of A.

13. A square matrix A has inverse if and only if A is non-singular.

14. Unique solution of equation AX = B is given by X = A–1 B, where A 0 ≠ .

15. A system of equation is consistent or inconsistent according as its solution exists or not.

16. For a square matrix A in matrix equation AX = B (i) |A| ≠ 0, there exists unique solution (ii) |A| = 0 and (adj A) B ≠ 0, then there exists no solution (iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent.