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Matrix Multiplication - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Multiplication of two matrices, Properties of Matrix Multiplication is considered one of the most asked concept.

  • 47 Questions around this concept.

Solve by difficulty

 If  then which one of the following statements is not correct ?

If  then adj \left ( 3A^{2} +12A\right )  is equal to :

 

Let   .The only correct statement about the matrix A is

If A and B are square matrices of size n x n such that A^{2}-B^{2}=(A-B)(A+B), then which of the following will be always true?

Let If u1 and u are column matrices such that

 then  u1 + u2 is equal to :

Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$

Given below are two statements :
Statement I : $f(-x)$ is the inverse of the matrix $f(x)$.
Statement II : $\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})=\mathrm{f}(\mathrm{x}+\mathrm{y})$.

In the light of the above statements, choose the correct answer from the options given below:

 

Concepts Covered - 2

Multiplication of two matrices

Matrix multiplication: 

Product AB can be found if the number of columns in matrix A and the number of rows in matrix B are equal. Otherwise, multiplication AB is not possible.

i) AB is defined only if col(A) = row(B)

ii) BA is defined only if col(B) = row(A)

If 

    \\\mathrm{A = \left [ a_{ij} \right ]_{m\times n}} \\\mathrm{\\B=\left [ b_{ij} \right ]_{n\times p}}

    \\\mathrm{C = AB = \left [ c_{ij} \right ]_{m\times p}} \\\mathrm{Where\;\; c_{ij} = \sum_{j=1}^{n}a_{ij}b_{jk}, 1\leq i\leq m,1\leq k\leq p} \\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=a_{i1}b_{1k} + a_{i2}b_{2k} + a_{i3}b_{3k}+ ... + a_{in}b_{nk}}

For examples

\\\mathrm{Suppose,\;two\;matrices\;are\;given}\\\mathrm{A=\begin{bmatrix} a_{11} &a_{12} &a_{13} \\ a_{21} &a_{22} & a_{33} \end{bmatrix}_{2\times3}\;\;\;and\;\;\;B=\begin{bmatrix} b_{11}& b_{12} &b_{13} \\b_{21} &b_{22} &b_{23} \\b_{31} &b_{32} &b_{33} \end{bmatrix}_{3\times3}}\\\\\mathrm{To\:obtain\:the\:entries\:in\:row\:\mathit{i}\,and\,column\,j\,\:of\:AB,\:we\:multiply\:the\:entries\:in\:row\:\mathit{i}\:of\:A\:by\:}\\\mathrm{column\:\mathit{j}\:in\:B\:and\:add.}\\\mathrm{given\:matrices\:A\:and\:B,\:where\:the\:order\:of\:A\:are\:2\times3\:and\:the\:order\:of\:B\:are\:3\times3,}\\\mathrm{the\:product\:of\:AB\:will\:be\:a\:2\times3\:matrix.}\\\\\mathrm{To\:obtain\:the\:entry\:in\:row\:1,\:column\:1\:of\:AB,\:multiply\:the\:}\\\mathrm{first\:row\:in\:A\:by\:the\:first\:column\:in\:B,and\:add.}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{bmatrix} a_{11} &a_{12} &a_{13} \end{bmatrix}\begin{bmatrix} b_{11}\\b_{21} \\b_{31} \end{bmatrix}=a_{11}\cdot b_{11}+a_{12}\cdot b_{21}+a_{13}\cdot b_{31}}

\\\mathrm{To\:obtain\:the\:entry\:in\:row\:1,\:column\:2\:of\:AB,\:multiply\:the\:}\\\mathrm{first\:row\:in\:A\:by\:the\:second\:column\:in\:B,and\:add.}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{bmatrix} a_{11} &a_{12} &a_{13} \end{bmatrix}\begin{bmatrix} b_{12}\\b_{22} \\b_{32} \end{bmatrix}=a_{11}\cdot b_{12}+a_{12}\cdot b_{22}+a_{13}\cdot b_{32}}\\\\\mathrm{To\:obtain\:the\:entry\:in\:row\:1,\:column\:3\:of\:AB,\:multiply\:the\:}\\\mathrm{first\:row\:in\:A\:by\:the\:third\:column\:in\:B,and\:add.}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{bmatrix} a_{11} &a_{12} &a_{13} \end{bmatrix}\begin{bmatrix} b_{13}\\b_{23} \\b_{33} \end{bmatrix}=a_{11}\cdot b_{13}+a_{12}\cdot b_{23}+a_{13}\cdot b_{33}}\\\\\mathrm{We\:proceed\:the\:same\:way\:to\:obtain\:the\:second\:row\:of\:AB.\:In\:other\:words,\:}\\\mathrm{row\:2\:of\:A\:times\:column\:1\:of\:B;}\\\mathrm{row\:2\:of\:A\:times\:column\:2\:of\:B;}\\\mathrm{row\:2\:of\:A\;times\:column\:3\:of\:B.}

\\\mathrm{When\:complete,\:the\:product\:matrix\:will\:be}\\\\\mathrm{AB=\begin{bmatrix} a_{11}\cdot b_{11}+a_{12}\cdot b_{21}+a_{13}\cdot b_{31}\;\;& a_{11}\cdot b_{12}+a_{12}\cdot b_{22}+a_{13}\cdot b_{32}\;\; &a_{11}\cdot b_{13}+a_{12}\cdot b_{23}+a_{13}\cdot b_{33} \\ a_{21}\cdot b_{11}+a_{22}\cdot b_{21}+a_{23}\cdot b_{31} \;\;& a_{21}\cdot b_{12}+a_{22}\cdot b_{22}+a_{23}\cdot b_{32} \;\;& a_{21}\cdot b_{13}+a_{22}\cdot b_{23}+a_{23}\cdot b_{33} \end{bmatrix}}

 

Properties of Matrix Multiplication

Properties of matrix multiplication:

    i) Multiplication may or may not be commutative, so AB may or may not be equal to BA.    

    ii) Matrix multiplication is associative, meaning A(BC) = (AB)C

    iii) Matrix multiplication is distributive over addition, mean  A(B+C) = AB + AC  and (B+C)A = BA + CA

    iv) If matrix multiplication of two matrices gives null matrix then it doesn’t mean that any of those two matrices was a null matrix.

             So         AB = O ⇏ A = O or B = O.

           A=\left[\begin{array}{ll} 0 & 2 \\ 0 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right], \text { then } A B=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]

    v) Cancellation law in matrix multiplication doesn’t hold, means  AB = AC ⇏ B = C

    vi) Matrix multiplication A x A is represented by A2. Thus, A・A・A・A……...n times = An.

    vii) if A is m x n matrix then, \mathrm{I_{m} A=A=A I_{n}}.

Study it with Videos

Multiplication of two matrices
Properties of Matrix Multiplication

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