Matrices

1. Equal matrices
   Two matrices A and B are called equal iff
   (i) A, B are of the same order and (ii) their corresponding entries are equal.
   If A = and B = , then A = B iff a = p, b = q, c = r and d = s.
2. Addition of matrices
   If A and B are two matrices of the same order then A +B is the matrix obtained by adding the corresponding elements of A and B.
   If A = and B = , then A +B =
3. Subtraction of matrices.
   If A and B are two matrices of the same order then A -B is the matrix obtained by subtracting the elements of B from the corresponding elements of A.
   If A = and B = , then A -B =
4. Multiplication by a number
   If k is any number and A is a matrix, then kA is a matrix obtained by multiplying each element of the matrix A by k.
   If A = , then kA = .
5. Multiplication of two matrices.
   Two matrices A and B are said to be conformable for the product AB iff the number of columns in A is equal to the number of rows in B.
6. If A is of order m × n and B is of order n × p, then AB is of order m × p, and is defined as AB = [C_ik]_mxp where (i, k) th element of AB = sum of the products of the elements of the ith row of A with the corresponding element of the kth column of B.
   AB =

Exercise

1. Find the values of a, b, c and d if
2. Find the values of a and b if
3. If , find the values of x, y, a and b.
4. If , find the values of x, y, a and b.
5. If A = , B = and C = , find
   (i) A -B (ii) 3A -C (iii) AB (iv) BA (v) CB
6. If A = , B = and C = , find A +2B -3C.
7. Let A = , B = and C = .
   (i) Verify that (A +B) +C = A +(B +C).
   (ii) Find 2A -3B +4C.
8. , find the values of x, y and z.
9. Find X if Y = and 2X +Y = .
10. Let A = and B = , find the matrix X if 2A +3X = 5B.
11. If A = , B = and C = , verify that
    (i) (AB)C = A(BC)
    (ii) A(B +C) = AB +AC
    (iii) (A -B)C = AC -BC
12. If A = , prove that (A -2I)(A -3I) = O where I is the unit matrix of order 2.
13. If A = and B = , find A (BA).
14. If A = , B = and C = , show that AB = AC.
15. If A = , B = and C = , find A(BC) and A(CB). Is A(BC) = A(CB)?
16. If A = and B = , does (A +B)² = A² +2 AB +B² hold?
17. If A = , show that A² -3I = 2A, where I is the unit matrix of order 2.
18. If , find the values of a, b and c.
19. If A = and B = , find the value of x given that A² = B.
20. If A = , B = and AB = A +B, find the values of a, b and c.

Answers

1. a = 4, b = 5, c = 2, d = -3     2. a = 2, b = 2
3. x = 1, y = 2, a = 3, b = 4       4. x = 2, y = 1, a = -1, b = 0
5. (i)   (ii)    (iii)
    (iv) (v)
  6.                     7. (ii)
  8. x = 2, y = -8, z = 7      9.
10.                  13.
15. ; A(BC) A(CB)        16. No
18. a = 3, b = 9, c = 3      19. 12     20. a = 3/2, b = 0, c = 4/3