Matrix Algebra:

Matrix has emerged as a great mathematical tool which simplifies our work to a great extent. As we know only straight long methods of calculation but this mathematics tool made it easy. By the emergence of concept of matrix algebra, we can obtain compact and simple methods of solving system of linear equations and other algebraic calculation.

Simply matrix algebra is a puzzle game. You must enjoy playing it. It is the different type of arrangement of numbers, symbols or expression in several rows and columns. Or by definition, it is said that a matrix is an ordered rectangular array of numbers or functions.

Let’s see the example:
\(\begin{bmatrix} 2 & 5 & 1\\ 7 & 9 & 3\\ -4 & 5 & 6 \end{bmatrix}\)

Rows: The horizontal lines from left to right in the above matrix is said to be rows.

Columns: Then vertical lines from up to down in the above matrix is said to be columns.

Order of matrix:

A matrix which has m rows and n columns. We say this type of matrix as matrix of order m × n.

We can express the order of any matrix as:
A =\( [a_{ij}]_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & … & a_{1n}\\ a_{21} & a_{22} & … & a_{2n}\\ .& .& … &. \\ .& . & … &. \\ a_{m1} & a_{m2} & … & a_{mn} \end{bmatrix} _{m \times n}\)

Also note that  1 ≤ I ≤ m,1 ≤ j ≤ n also i, j ∈ N

Types of matrix:

Addition and Subtraction of Matrices:

In matrix algebra the addition and subtraction of any two matrix is only possible when both the matrix is of same order.

Addition: There is addition law for matrix addition. You should only add the element of one matrix to the corresponding elements only. i.e a_ij + b_ij = c_ij

\(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) + \(\begin{bmatrix} e & f \\ g & h \end{bmatrix}\) = \(\begin{bmatrix} a+e & b + f \\ c + g & d + h \end{bmatrix}\)

Subtraction: There is also subtraction law for matrix addition. You should only add the element of one matrix to the corresponding elements only. i.e a_ij – b_ij = d_ij

\(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) – \(\begin{bmatrix} e & f \\ g & h \end{bmatrix}\) = \(\begin{bmatrix} a – e & b – f \\ c – g & d – h \end{bmatrix}\)

Matrix multiplication:

Matrix algebra for multiplication are of two types:

1. Scalar multiplication: we may define multiplication of a matrix by a scalar as follows:
   if A = [a_ij] m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.
2. Vector Multiplication: Two matrices A and B can only be multiplied if and only if the number of column of matrix A is equal to the number of rows of matrix B or vice versa.

Note: A_m×n × B_p×q = M_m×q

We can understand matrix multiplication by following rule:

\(\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}_{3 \times 3} \times \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33} \end{bmatrix}_{3 \times 3}\)

=\(\begin{bmatrix} (a_{11}\times b_{11} + a_{12}\times b_{21}+ a_{13}\times b_{31}) & (a_{11}\times b_{12} + a_{12}\times b_{22}+ a_{13}\times b_{32}) & (a_{11}\times b_{13} + a_{12}\times b_{23}+ a_{13}\times b_{33}) \\ (a_{21}\times b_{11} + a_{22}\times b_{21}+ a_{23}\times b_{31}) & (a_{21}\times b_{12} + a_{22}\times b_{22}+ a_{23}\times b_{32}) & (a_{21}\times b_{13} + a_{22}\times b_{23}+ a_{23}\times b_{33})\\ (a_{31}\times b_{11} + a_{32}\times b_{21}+ a_{33}\times b_{31}) & (a_{31}\times b_{12} + a_{32}\times b_{22}+ a_{33}\times b_{32}) & (a_{31}\times b_{13} + a_{32}\times b_{23}+ a_{33}\times b_{33}) \end{bmatrix}_{3 \times 3}\)

Properties of matrix algebra:

Let two independent matrix in matrix algebra be A & B then,

1. A = [a_ij] = [b_ij] = B this is only possible if
   (i) A and B are of same order,
   (ii) a_ij = b_ij for all possible values of i and j.
2. kA = k[a_ij]_{m × n} = [k(a_ij)]_{m × n}
3. Negative of a matrix: – A = (–1)A
4. A – B = A + (–1) B
5. Matrix commutativity: A + B = B + A
6. Matrix assosiativity: (A + B) + C = A + (B + C), where A, B and C are of same order.
7. k(A + B) = kA + kB, where A and B are of same order, k is constant.
8. (k + l) A = kA + lA, where k and l are constant.
9. Matrix multiplication: (i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC

Transpose of matrix:

If A = [a_ij] _{m × n} , then A′ or A^T = [a_ji] _{n × m}

Properties of Transpose of matrix:

• (A′)′ = A,
• (kA)′ = kA′,
• (A + B)′ = A′ + B′,
• (AB)′ = B′A′

Types of Matrix as transpose:

1. Symmetric matrix:
   A is a symmetric matrix only if A′ = A.
2. Skew Symmetric Matrix:
   A is a skew-symmetric matrix only if A′ = –A.
   Note: Any square matrix can be represented as the sum of a symmetric and a skew-symmetric matrix.

Inverse of a matrix:

If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A.

Inverse of matrix A is denoted by A^–1 and A is the inverse of B.

Inverse of a square matrix, if it exists, is always unique. This is a great factor dealing with matrix algebra.

It is given that A^-1 = \(\frac{adj\: A}{|A|}\)

Matrix algebra problems:

1. Add the following matrix:
   \(\begin{bmatrix} 1 & 4\\ 2 & 9\\ 6 & 11 \end{bmatrix}_{3 \times 2}\) + \(\begin{bmatrix} 2 & 5\\ 7 & 16\\ 9 & 17 \end{bmatrix}_{3 \times 2}\)

Solution:
As the number of rows and column of first matrix is equal to the number of rows and columns  of the second matrix. Therefore,by matrix algebra the matrix addition is possible.

\(\begin{bmatrix} 1 & 4\\ 2 & 9\\ 6 & 11 \end{bmatrix}_{3 \times 2}\) + \(\begin{bmatrix} 2 & 5\\ 7 & 16\\ 9 & 17 \end{bmatrix}_{3 \times 2}\)
= \(\begin{bmatrix} 3 & 9\\ 9 & 25\\ 15 & 28 \end{bmatrix}_{3 \times 2}\)

We added all the corresponding elements.