# Matrices

*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.*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 =*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 =*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 = .*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.- 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}_{mx}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._{p}

AB =

## Exercise

- Find the values of a, b, c and d if
- Find the values of a and b if
- If , find the values of x, y, a and b.
- If , find the values of x, y, a and b.
- If A = ,
B =
and C = , find

(i) A -B (ii) 3A -C (iii) AB (iv) BA (v) CB - If A = , B = and C = , find A +2B -3C.
- Let A = ,
B =
and C = .

(i) Verify that (A +B) +C = A +(B +C).

(ii) Find 2A -3B +4C. - , find the values of x, y and z.
- Find X if Y = and 2X +Y = .
- Let A = and B = , find the matrix X if 2A +3X = 5B.
- If A = ,
B =
and C = , verify that

(i) (AB)C = A(BC)

(ii) A(B +C) = AB +AC

(iii) (A -B)C = AC -BC - If A = , prove that (A -2I)(A -3I) = O where I is the unit matrix of order 2.
- If A = and B = , find A (BA).
- If A = , B = and C = , show that AB = AC.
- If A = , B = and C = , find A(BC) and A(CB). Is A(BC) = A(CB)?
- If A = and B = , does (A +B)² = A² +2 AB +B² hold?
- If A = , show that A² -3I = 2A, where I is the unit matrix of order 2.
- If , find the values of a, b and c.
- If A = and B = , find the value of x given that A² = B.
- 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