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**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.