Matrix operations - Practice Questions & MCQ

Addition of matrices:

Two matrices can be added only when they are of the same order

If two matrices of A and B of the same order, they are said to be conformable for addition.

If A and B are matrices of order m × n, then their sum will also be a matrix of the same order and in addition, corresponding elements of A and B get added.

So if    for all i, j

eg

Subtraction of matrices:

Two matrices can be subtracted only when they are of the same order. If A and B are matrices of order m × n then their difference will also be a matrix of the same order and in subtraction, corresponding elements of A and B get subtracted. So if 

  for all i, j

eg

Properties of matrix addition:

1. Matrix addition is commutative, A + B = B + A

2. Matrix addition is associative, A + (B+C) = (A+B) + C

3. Additive identity exist, means there exist a matrix O (null matrix) such that  A + O = A = O + A (Here O has same order as A)

4. Existence of additive inverse means there exists a matrix B such that A + B = O = B + A

5. Cancellation property: 

      If A + B = A + C then B = C

      If A + C = B + C then A = B

Note: all matrix taken in above property explanation has same order which is m × n.

Scalar multiplication:

Let k be any scalar number, and   be a matrix. Then the matrix obtained by multiplying every element A by a scalar k and denoted as kA.

Example, if 

Properties of scalar multiplication: 

If A and B are two matrices and k, l are scalar then

    i) k(A + B)=kA + kB

    ii) kl(A) = k(lA) = l(kA)

    iii) (k+l)A = kA + lA

    iv) (-k) A = -(kA) = k(-A)

    v) 1A = A, (-1)A = -A

Note: A and B have the same order m × n.