Matrices

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Matrices – Summary
A matrix is an ordered rectangular array of numbers or functions.
A matrix having m and n columns is called a matrix of order m × n.
{[a_{ij}]}_{m × 1} is a column matrix.
{[a_{ij}]}_{1 × n} is a row matrix.
{\text{A} = {[a_{ij}]}_{m × n}} is a diagonal matrix if {a_{ij} = 0}, when ij.

{\text{A} = {[a_{ij}]}_{m × n}} is a scalar matrix if {a_{ij} = k}, (k is some constant), when {i = j}.
{\text{A} = {[a_{ij}]}_{m × n}} is an identity matrix, if {[a_{ij}] = 1}, when {i = j}, {[a_{ij}] = 0}, when ij
A zero matrix has all its elements as zero.
{\text{A} = [a_{ij}] = [b_{ij}] = \text{B}} if
(i)
A and B are of same order,
(ii)
{a_{ij} = b_{ij}} for all possible values of i and j.
{k\text{A} = k{[a_{ij}]}_{m × n} = [k(a_{ij})]_{m × n}}

{-\text{A} = (-1)\text{A}}
{\text{A} - \text{B} = \text{A} + (-1)\text{B}}
{\text{A} + \text{B} = \text{B} + \text{A}}
{(\text{A} + \text{B} + \text{C}) = \text{A} + (\text{B} + \text{C})}, where A, B and C are of the same order.
{k(\text{A} + \text{B}) = k\text{A} + k\text{B}}, where A and B are of same order, k is constant.

{(k + l)\text{A} = k\text{A} + l\text{A}}, where k and l are constants.
If {\text{A} = [a_{ij}]_{m × n}} and {\text{B} = {[b_{jk}]}_{n × p}} , then {\text{AB} = \text{C} = {[c_{ik}]}_{m × p}} , where {c_{ik} = \displaystyle\sum\limits_{j=1}^n a_{ij} b_{jk}}.
(i)
{\text{A}(\text{BC}) = (\text{AB})\text{C}},
(ii)
{\text{A}(\text{B} + \text{C}) = \text{AB} + \text{AC}},
(iii)
{(\text{A} + \text{B})\text{C}) = \text{AC} + \text{BC}},
If {\text{A} = {[a_{ij}]}_{m × n}} , then {\text{A}′} or {\text{A}^T = {[a_{ji}]}_{n × m}}
(i)
{(\text{A}′)′ = \text{A}},
(ii)
{(k\text{A})′ = k\text{A}′}
(iii)
{(\text{A} + \text{B})′ = (\text{A}′ + \text{B}′)}
(iii)
{(\text{AB})′ = (\text{B}′\text{A}′)}

\text{A} is a symmetric matrix if {\text{A}′ = \text{A}}
\text{A} is a skew matrix if {\text{A}′ = -\text{A}}
Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.
Elementary operations of a matrix are as follows:
(i)
{\text{R}_\text{i} ↔ \text{R}_\text{j}} or \text{C}_\text{i} ↔ \text{C}_\text{j}
(ii)
{\text{R}_\text{i} ↔ k\text{R}_\text{i}} or \text{C}_\text{i} ↔ k\text{C}_\text{j}
(iii)
{\text{R}_\text{i} ↔ \text{R}_\text{i} + k\text{R}_\text{j}} or {\text{C}_\text{i} ↔ \text{C}_\text{i} + k\text{C}_\text{j}}
If A and B are two square matrices such that {\text{AB} = \text{BA} = \text{I}}, then B is the inverse matrix of A and is denoted by {\text{A}^{-1}} and A is the inverse of B.
Inverse of a square matrix, if it exists, is unique.