This page contains the NCERT mathematics class 12 chapter Matrices Chapter Summary. You can find the summary for the chapter 3 of NCERT class 12 mathematics in this page. So is the case if you are looking for NCERT class 12 Maths related topic Matrices. This page contains summary of the chapter. If you’re looking for exercise solutions, they’re available at
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Matrices – Summary
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A matrix is an ordered rectangular array of numbers or functions.
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A matrix having m and n columns is called a matrix of order m × n.
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{[a_{ij}]}_{m × 1} is a column matrix.
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{[a_{ij}]}_{1 × n} is a row matrix.
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{\text{A} = {[a_{ij}]}_{m × n}} is a diagonal matrix if {a_{ij} = 0}, when i ≠ j.
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{\text{A} = {[a_{ij}]}_{m × n}} is a scalar matrix if {a_{ij} = k}, (k is some constant), when {i = j}.
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{\text{A} = {[a_{ij}]}_{m × n}} is an identity matrix, if {[a_{ij}] = 1}, when {i = j}, {[a_{ij}] = 0}, when i ≠ j
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A zero matrix has all its elements as zero.
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{\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.
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{k\text{A} = k{[a_{ij}]}_{m × n} = [k(a_{ij})]_{m × n}}
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{-\text{A} = (-1)\text{A}}
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{\text{A} - \text{B} = \text{A} + (-1)\text{B}}
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{\text{A} + \text{B} = \text{B} + \text{A}}
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{(\text{A} + \text{B} + \text{C}) = \text{A} + (\text{B} + \text{C})}, where A, B and C are of the same order.
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{k(\text{A} + \text{B}) = k\text{A} + k\text{B}}, where A and B are of same order, k is constant.
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{(k + l)\text{A} = k\text{A} + l\text{A}}, where k and l are constants.
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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}}.
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(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}},
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If {\text{A} = {[a_{ij}]}_{m × n}} , then {\text{A}′} or {\text{A}^T = {[a_{ji}]}_{n × m}}
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(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}′)}
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\text{A} is a symmetric matrix if {\text{A}′ = \text{A}}
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\text{A} is a skew matrix if {\text{A}′ = -\text{A}}
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Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.
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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}}
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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.
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Inverse of a square matrix, if it exists, is unique.