\n
1. Equality of Matrices<\/h4>\n
Two matrices $A = (a_{jk})$ and $B = (b_{jk})$ of the same order [i.e. equal numbers of rows and columns] are equal<\/em> if and only if $a_{jk} = b_{jk}$ . <\/p>\n
2. Addition of Matrices<\/h4>\n
If $A = (a_{jk})$ and $B = (b_{jk})$ have the same order we define the sum<\/em> of $A$ and $B$ as $A + B = (a_{jk} + b_{jk})$ . <\/p>\n
Note that the communicative and associative laws for addition are satisfied by matrices, i.e. for any matrices $A,\\ B,\\ C$ of the same order <\/p>\n
$A + B = B + A,\\ A + (B + C) = (A + B) + C \\cdots (2)$ <\/p>\n
3. Subtraction of Matrices<\/h4>\n
If $A = (a_{jk})$ , $B = (b_{jk})$ have the same order, we define the difference<\/em> of $A$ and $B$ as $A - B = (a_{jk} - b_{jk})$ . <\/p>\n
4. Multiplication of a Matrix by a Number<\/h4>\n
If $A = (a_{jk})$ and $\\lambda$ is any number or scalar, we define the product<\/em> of $A$ by $\\lambda$ as $\\lambda A = A\\lambda = (\\lambda a_{jk})$ . <\/p>\n
5. Multiplication of Matrices<\/h4>\n
If $A = (a_{jk})$ is an $m\\times n$ matrix while $B = (b_{jk})$ is an $n\\times p$ matrix, then we define the product<\/em> $A\\cdot B$ or $AB$ as the matrix $C = (c_{jk})$ where <\/p>\n
$\\displaystyle c_{jk} = \\sum_{l = 1}^n a_{jl}b_{lk} \\cdots (3)$ <\/p>\n
and where $C$ is of order $m\\times p$ . <\/p>\n
Note that in general $AB \\ne BA$ , i.e. the communicative law for multiplication of matrices is not satisfied in general. However, the associative and distributive laws are satisfied, i.e. <\/p>\n
$A(BC) = (AB)C,\\ A(B + C) = AB + AC,\\ (B + C)A = BA + CA \\cdots (4)$ <\/p>\n
A matrix $A$ can be multiplied by itself if and only if it is a square matrix. The product $A\\cdot A$ can in such case be written $A^2$ . Similarly we define powers of a square matrix, i.e. $A^3 = A\\cdot A^2,\\ A^4 = A\\cdot A^3$ , etc. <\/p>\n
6. Transpose of a Matrix<\/h4>\n
If we interchange rows and columns of a matrix $A$ , the resulting matrix is called the transpose<\/em> of $A$ and is denoted by $A^T$ . In symbols, if $A = (a_{jk})$ then $A^T = (a_{kj})$ . <\/p>\n
We can prove that <\/p>\n
$(A + B)^T = A^T + B^T,\\ (AB)^T = B^TA^T,\\ (A^T)^T = A \\cdots(5)$ <\/p>\n
7. Symmetric and Skew-Symmetric matrices<\/h4>\n
A square matrix $A$ is called symmetric<\/em> if $A^T = A$ and skew-symmetric<\/em> if $A^T = - A$ . <\/p>\n
Any real square matrix [i.e. one having only real elements] can always be expressed as the sum of a real symmetric matrix and a real skew-symmetric matrix. <\/p>\n
8. Complex Conjugate of a Matrix<\/h4>\n
If all elements $a_{jk}$ of a matrix $A$ are replaced by their complex conjugates $\\bar{a}_{jk}$ , the matrix obtained is called the complex conjugate<\/em> of $A$ and is denoted by $\\bar{A}$ . <\/p>\n
9. Hermitian and Skew-Hermitian Matrices<\/h4>\n
A square matrix $A$ which is the same as the complex conjugate of its transpose, i.e. if $A = \\bar{A}^T$ , is called Hermitian<\/em>. If $A = -\\bar{A}^T$ , then $A$ is called skew-Hermitian<\/em>. If $A$ is real these reduce to symmetric and skew-symmetric matrices respectively. <\/p>\n
10. Principal Diagonal and Trace of a Matrix<\/h4>\n
If $A = (a_{jk})$ is a square matrix, then the diagonal which contains all elements $a_{jk}$ for which $j = k$ is called the principal<\/em> or main diagonal<\/em> and the sum of all elements is called trace<\/em> of $A$ . <\/p>\n
A matrix for which $a_{jk} = 0$ when $j \\ne k$ is called diagonal matrix<\/em>. <\/p>\n
11. Unit Matrix<\/h4>\n
A square matrix in which all elements of the principal diagonal are equal to 1 while all other elements are zero is called the unit matrix<\/em> and is denoted by $I$ . An important property of $I$ is that <\/p>\n
$AI = IA = A,\\ I^n = I,\\ n = 1,2,3,\\cdots(6)$ <\/p>\n
The unit matrix plays a role in matrix algebra similar to that played by the number one in ordinary algebra. <\/p>\n
12. Zero or Null matrix<\/h4>\n
A matrix whose elements are all equal to zero is called the null<\/em> or zero matrix<\/em> and is often denoted by $O$ or symply 0. For any matrix $A$ having the same order as 0 we have <\/p>\n
$A + 0 = 0 + A = A \\cdots(7)$ <\/p>\n
Also if $A$ and 0 are square matrices, then<\/p>\n
$A0 = 0A = 0 \\cdots(8)$ <\/p>\n
The zero matrix plays a role in matrix algebra similar to that played by the number zero of ordinary algebra. <\/p>\n<\/blockquote>\n
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1. Equality of Matrices<\/h4>\nTwo matrices and of the same order [i.e. equal numbers of rows and columns] are equal<\/em> if and only if . <\/p>\n

1. Equality of Matrices<\/h4>\n
Two matrices $A = (a_{jk})$ and $B = (b_{jk})$ of the same order [i.e. equal numbers of rows and columns] are equal<\/em> if and only if $a_{jk} = b_{jk}$ . <\/p>\n