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1. Equality of Two Matrices:

If A = [a_{ij}] and B = [b_{ij}]are the two matrices of same order then they are said to be equal if their corresponding elements are same. i.e., A = B if a_{ij} =b_{ij} for all i and j.

2. Addition and Subtraction of Two Matrices:

If A = [a_{ij}]_{m}_{ ´ }_{n} and B = [b_{ij}]_{m}_{ ´ }_{n} then A + B = [a_{ij} + b_{ij}]_{m}_{ ´ }_{n} and A – B= [a_{ij} – b_{ij}]_{m}_{ ´ }_{n} for all i and j.

3. If A, B, C are three matrices of same order, then

(i)

A + B = B + A, Commutative law.

(ii)

A + (B + C) = (A + B) + C, Associative law

(iii)

A + 0 = 0 + A = A, where 0 is a null matrix

4. Multiplication of two Matrices:

If A = [a_{ij}]_{m}_{ ´ }_{n} and B = [b_{jk}]_{n}_{ ´ }_{p} are two matrices then the product C = AB = [C_{ik}]_{m}_{ ´ }_{n} where C_{ik} can be obtained by multiplying the i^{th} row of A and k^{th} column of B.

5. Properties of Matrix Multiplication :

If A, B, C are three matrices, then

·

AB ¹ BA (in general)

A(BC) = (AB)C, associative law.

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

distributive law.

A.0 = 0.A = 0, where 0 is a null matrix.

A.I = I.A = A, where I is an identity matrix.

6. Properties of Transpose :

If A and B are two matrices and k is a scalar, then

7. Some useful results:

8. Some useful properties: