Operations On Matrices
1) MATRIX ADDITION:
The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrix.
If is a 2 x 2 matrix and is another 2 x 2 matrix, then
Or
If A = [a_{ij}] and B = [b_{ij}] then,
A + B = [a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}] for all i and j.
Example: Given and , find A + B.
Solution:
If is a 2 x 2 matrix and is another 2 x 2 matrix, then
Or
If A = [a_{ij}] and B = [b_{ij}] then,
A + B = [a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}] for all i and j.
Example: Given and , find A + B.
Solution:
PROPERTIES OF MATRIX ADDITION:
- Commutative Property: If A = [a_{ij}] and B = [b_{ij}] are two matrices of same order, then A + B = B + A.
- Associative Property: If A = [a_{ij}] and B = [b_{ij}] and C = [c_{ij}] are matrices of same order, then (A + B) + C = A + (B + C).
- Additive Identity: If A = [a_{ij}] be a matrix of order mx n and O be a zero matrix of order mx n, then A + O = O + A = A. Hence the matrix O is the additive identity for matrix addition.
- Additive Inverse: For every matrix A = [a_{ij}]_{mxn} , there exists a matrix -A = [-a_{ij}]_{mxn} such that A + (-A) = 0. The matrix -A is known as the additive inverse of A.
2) MATRIX MULTIPLICATION:
MULTIPLICATION OF A MATRIX BY A SCALAR:
If A = [a_{ij}]_{mxn} is a matrix and k is a scalar, then k.A is another
matrix which is obtained by multiplying each element of A by the scalar k, that is,
k.A = k[a_{ij}]_{mxn} = [k(a_{ij})]_{mxn} which means that (i, j)th element of kA is ka_{ij}.
Example:
If then find 2A.
Solution:
k.A = k[a_{ij}]_{mxn} = [k(a_{ij})]_{mxn} which means that (i, j)th element of kA is ka_{ij}.
Example:
If then find 2A.
Solution:
MULTIPLICATION OF MATRICES:
Two matrices A and B can be multiplied only if the number of columns of A is equal
to the number of rows of B. Hence, if A is a m x n matrix, then to find the product of
A with a matrix B, the matrix B must have n rows and any finite number of columns.
Let A be a matrix of order m x n and B is a matrix of order n x p, then the product matrix C of matrices A and B will be:
A x B = C
Hence the order of the product matrix C will be m x p.
To find the (i, j)th element of C which is c_{ij}, the ith row of A and jth column of B are taken and multiplied (element wise) and then the sum of all these products is obtained.
Hence, ith row of A = [a_{i1} a_{i2} a_{i3} .... a_{in}]
And the jth column of
Hence c_{ij} = a_{i1}. b_{1j} + a_{i2}. b_{2j} + a_{i3}. b_{3j} + .... + a_{in}. b_{nj}
=
Example:
If and , then find AB.
Solution:
Hence
PROPERTIES OF MULTIPLICATION OF MATRICES:
3) ELEMENTARY OPERATIONS ON A MATRIX:
There are 3 operations which can be performed on matrices. Three of these operations are due to rows and three are due to columns. These are known as elementary operations or transformations.
Let A be a matrix of order m x n and B is a matrix of order n x p, then the product matrix C of matrices A and B will be:
A x B = C
Hence the order of the product matrix C will be m x p.
To find the (i, j)th element of C which is c_{ij}, the ith row of A and jth column of B are taken and multiplied (element wise) and then the sum of all these products is obtained.
Hence, ith row of A = [a_{i1} a_{i2} a_{i3} .... a_{in}]
And the jth column of
Hence c_{ij} = a_{i1}. b_{1j} + a_{i2}. b_{2j} + a_{i3}. b_{3j} + .... + a_{in}. b_{nj}
=
Example:
If and , then find AB.
Solution:
Hence
PROPERTIES OF MULTIPLICATION OF MATRICES:
- Associative Property: If A, B and C are any three compatible matrices, then A(BC) = (AB)C.
- Distributive Property: If A, B and C are any three compatible matrices, then
- A(B + C) = AB + BC
- (A + B)C = AC + BC
- Multiplicative Identity: For every square matrix A, there exists an identity matrix of the same order such that IA = AI =A.
3) ELEMENTARY OPERATIONS ON A MATRIX:
There are 3 operations which can be performed on matrices. Three of these operations are due to rows and three are due to columns. These are known as elementary operations or transformations.
- Interchange of any two rows or columns: If ith row (column) of a matrix is interchanged with the jth row (column), then it is represented by R_{i} ↔ R_{j} (C_{i} ↔ C_{j}).
- The multiplication of the elements of any row or column by a non-zero scalar: If the elements of ith row (column) are multiplied by a non-zero scalar, then it is represented byR_{i} → k.R_{i} (C_{i} → k.C_{i}).
- The addition to the element of any row (column), the corresponding elements of any other row or column multiplied by any non-zero scalar. If k times the elements of jth row (column) are added to the corresponding elements of the ith row (column), then it is denoted by
R_{i} → R_{i} + k.R_{j} (C_{i} → C_{i}+ k.C_{i}).
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