## Algebra Of Matrices

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# Algebra of Matrices

Addition of Matrices. Let A and B be two matrices of the same order. Then the addition of A + B, is the matrix obtained by adding corresponding entries of A and B.

Thus, if A = (a

A + B = (a

Remark. Notice that we can add two matrices only if they are of the same order. If they are, we say they are conformable for addition, Also, the order of the sum of two matrices is same as that of the two original matrices.

Example.

Example. The addition of

is not defined since the two matrices are not of the same order.

If A is any matrix, the negative of A, denoted by – A, is the matrix obtained by replacing each entry in A by its negative. For example, if

1. Matrix addition is commutative. That is, if A and B be two matrices of the same order, then A+B = B+A.

2. Matrix addition is associative. That is, if A, B and C be three matrices of the same order, then (A+B) + C = A + (B+C).

3. Existence of additive identity. That is, if O is the zero matrix of the same order as that of the matrix

A, then A + O = A = O+A.

4. Existence of additive inverse. That is, if A be any matrix, then A + (-A) = O = (-A) + A

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Thus, if A = (a

_{ij})mxn and B = ( b_{ij})mxn, thenA + B = (a

_{ij}+ b_{ij})mxnRemark. Notice that we can add two matrices only if they are of the same order. If they are, we say they are conformable for addition, Also, the order of the sum of two matrices is same as that of the two original matrices.

Example.

Example. The addition of

is not defined since the two matrices are not of the same order.

If A is any matrix, the negative of A, denoted by – A, is the matrix obtained by replacing each entry in A by its negative. For example, if

**Properties of Addition of Matrices**1. Matrix addition is commutative. That is, if A and B be two matrices of the same order, then A+B = B+A.

2. Matrix addition is associative. That is, if A, B and C be three matrices of the same order, then (A+B) + C = A + (B+C).

3. Existence of additive identity. That is, if O is the zero matrix of the same order as that of the matrix

A, then A + O = A = O+A.

4. Existence of additive inverse. That is, if A be any matrix, then A + (-A) = O = (-A) + A

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