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Matrix operations mainly include four basic algebraic operations namely, the addition of matrices, subtraction of matrices, and multiplication of matrices and division of matrices. We all know that Matrix is an array of numbers or expressions arranged in rows (horizontal array) and columns (vertical array).

All these operations on the given matrix are covered along with their properties which also includes the elementary operations, i.e., row and column operations in detail which is given in the linked articles below. As discussed above addition, subtraction, multiplication, and division are the four basic operations on the matrix. To add or subtract matrices, these must be of the same order and for multiplication of matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

The order of the matrix is the number of rows and columns that a given matrix is having or in other words, the order can be defined as the dimension of the matrix. In any matrix, the number of rows is listed first and is followed by the number of columns. Thus, if any matrix has the order (or dimension) as 3 x 4, it means that it has 3 rows and 4 columns.

If two matrices A[a_{ij}]_{mxn} and B[b_{ij}]_{mxn} are of the same order then their sum A + B is a matrix of the order mxn. Each element of the resultant matrix is the sum of the corresponding elements of the matrices A and B, i.e., A + B = [a_{ij} + b_{ij}]_{mxn}.

\[\begin{bmatrix}1 &2 &3 \\4 &5 &6 \\7 &8 &9 \end{bmatrix} + \begin{bmatrix}9 &8 &7 \\6 &5 &4 \\3 &2 &1 \end{bmatrix} = \begin{bmatrix}1 + 9 &2 + 8 &3 + 7 \\4 + 6 &5 + 5 &6 + 4 \\7 + 3 &8 + 2 &9 +1 \end{bmatrix}\]

\[=\begin{bmatrix}10 &10 &10 \\10 &10 &10 \\10 &10 &10 \end{bmatrix}\]

Properties of Matrix Addition: If A, B, and C are three given matrices of same the order, then,

Commutative Law: Matrix addition is commutative, A + B = B + A.

Associative Law: Matrix addition is associative, (A + B) + C = A + (B + C).

Identity Matrix: Matrix addition follows identity rule, i.e., O + A = A = A + O, where O is zero matrix which is additive identity of the matrix.

Additive Inverse: A + (-A) = 0 = (-A) + A, where (-A) is obtained by changing the sign of every element of A which is additive inverse of the matrix.

If two matrices A[a_{ij}]_{mxn} and B[b_{ij}]_{mxn} are of the same order then their difference A - B is defined as a matrix of order mxn and is found by adding a negative of the corresponding elements of the second matrix to the first one.

A - B = A + (-B) = [a_{ij} - b_{ij}]_{mxn}

\[\begin{bmatrix}1 &2 &3 \\4 &5 &6 \\7 &8 &9 \end{bmatrix} + \begin{bmatrix}9 &8 &7 \\6 &5 &4 \\3 &2 &1 \end{bmatrix} = \begin{bmatrix}1 - 9 &2 - 8 &3 - 7 \\4 - 6 &5 - 5 &6 - 4 \\7 - 3 &8 - 2 &9 -1 \end{bmatrix}\]

\[=\begin{bmatrix}-8 &-6 &-4 \\-2 &0 &2 \\4 &6 &8 \end{bmatrix}\]

Multiplication of two matrices is defined only when the number of columns of the 1st matrix is equal to the number of rows of the 2nd matrix and the resultant or product will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.

Following are the examples of matrix multiplication:

i. Multiplication of 2 × 2 Matrices

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ii. Multiplication of 3 × 3 Matrices

\[\begin{bmatrix}a &b &c \\d &e &f \\g &h &i \end{bmatrix} \times \begin{bmatrix}j &k &l \\m &n &o \\p &q &r \end{bmatrix} = \begin{bmatrix}aj + bm + cp &ak + bn + cq &al + bo + cr \\dj + em + fp &dk + en + fq &dl + eo + fr \\gj + hm + ip &gk + hn + iq &gl + ho + ir \end{bmatrix}\]

Associative Law: Matrix multiplication follows associative law, i.e., for any three given matrices A, B, and C,

(AB) C = A (BC), whenever both sides of the multiplication or equality are defined.

Distributive Law: Matrix multiplication follows distributive law for three matrices, i.e., if A, B, and C are three matrices then,

A (B + C) =AB + AC

(A + B) C = AC + BC, whenever both sides of the equality are defined.

Existence of Multiplicative Identity: Matrix multiplication follows multiplicative identity, i.e., for every square matrix A, there exists an identity matrix of the same order such that IA = AI = A.

Properties of Scalar Multiplication of Matrices: If A and B are two matrices and m and n be any scalar:

m(A + B) = mA + mB

(m + n)A = mA + nA

m(nA) = mnA = n(mA)

Technically there is nothing like division of matrix, i.e., A÷B is undefined instead we write A × B-1 i.e product of A and B-1.

FAQ (Frequently Asked Questions)

Q1.What are the Various Forms of Matrix Operations?

Answer: There are three forms of the matrix operations defined, i.e., addition of matrices, subtraction of matrices, and multiplication of matrices. Technically the division of two matrices, A ÷ B is undefined, instead we write A × B^{-1} i.e product of A and B^{-1}.

Q2.How can we Multiply Matrices?

Answer: Multiplication of matrices can be done only when the number of columns in the 1st matrix equals the number of rows in the 2nd matrix and the resultant matrix thus obtained will be having the same number of rows as present in 1st matrix, and the same number of columns as that in the 2nd matrix.

Q3.What are the Various Forms of the Matrix?

Answer: The various forms of matrices are as follows.

**Column Matrix:**Matrix having only one column.**Row Matrix:**Matrix having only one row.**Square Matrix:**Matrix having an equal number of rows and columns.**Zero Matrix:**Matrix having all its elements equal to zero.**Upper Triangular Matrix:**Matrix having all elements below the main diagonal equal to zero.**Lower Triangular Matrix:**Matrix having all elements above the main diagonal equal to zero.**Diagonal Matrix:**Matrix having all elements apart from the main diagonal equal to zero.