Tag: maths

Quizzes (518) Questions (9324)

Questions Related to maths

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

 For what value of
k, the matrix $A = \begin{bmatrix} 4 & 3 -k\\ 1 & 2 \end{bmatrix}$ is
not invertible?

  1. (a)$k = -5,$
  2. (b)$k=5,$
  3. (c)$k=0$
  4. (d)$k=3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A matrix is not invertible if its determinant is zero. Det(A) = (4 * 2) - (1 * (3 - k)) = 8 - 3 + k = 5 + k. Setting 5 + k = 0 gives k = -5.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If the traces of $A, B$ are $20$ and $-8$, then the trace of $A+B$ is:

  1. $12$
  2. $-12$
  3. $28$
  4. $-28$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

 the trace of an $n\times n$ square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A

the traces of A,B are 20 and −8, then the trace of $A+B$ is $trac(A+B)=trace(A)+trac(B)=20+-8=12$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If$A=\left[ \begin{matrix} 1 & -5 & 7 \ 0 & 7 & 9 \ 11 & 8 & 9 \end{matrix} \right] $ , then  trace of matrix $A$ is

  1. $17$
  2. $25$
  3. $3$
  4. $12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $A=\left[ \begin{matrix} 1 & -5 & 7 \ 0 & 7 & 9 \ 11 & 8 & 9 \end{matrix} \right] $
$tr(A)=sum\ of\ diagonal\ matrix$
$tr(A)=1+7+9=17$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $\displaystyle :A= \left [ a _{ij} \right ]$ is a scalar matrix of order $\displaystyle :n\times n$ such that $\displaystyle :a _{ij}= k $ for all then trace of A is equal to

  1. $\displaystyle \:nk$
  2. $\displaystyle \:n+k$
  3. $\displaystyle \:n/k$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By definition of trace of a scalar matrix of order n, 
$tr(A)=a _{11}+ a _{22}+a _{33}+.....+a _{nn}$

$=k+k+....k $ (upto n times)

$tr(A)=nk$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $\displaystyle :A= \left [ a _{ij} \right ]$ is a scalar matrix, then trace of A is

  1. $\displaystyle \:\sum _{i} \sum _{i} a _{ij}$
  2. $\displaystyle \:\sum _{i} a _{ij}$
  3. $\: \sum _{ i } a _{ ij }\times { a } _{ ji }$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By definition of trace of a matrix of order n, 
$tr(A)=a _{11}+ a _{22}+a _{33}+.....+a _{nn}$
$\displaystyle =: \sum _{ i=j }  { a } _{ ij } =: \sum _{ i } a _{ ij }$
Hence, option 'B' is correct.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $A = \left[ {{a _{ij}}} \right]$ and ${a _{ij}} = i\left( {i + j} \right)$ then trace of $A=$

  1. $\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
  2. $\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{3}$
  3. $\frac{{n\left( {n + 1} \right)}}{2}$
  4. $\frac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $A=[a _{i,j}]$ and  $a _{i,j}=i(i+j)$ 

let the order of $A$ = $n\times n$

trace of $A$ =sum of all diagonal elements i.e.,$\sum[a _{i,j}]$ where $i=j$

$a _{1,1}=1(1+1)=2$
$a _{2,2}=2(2+2)=8$
$a _{3,3}=3(3+3)=18$
              $.$
              $.$
              $.$
$a _{n,n}=n(n+n)=2n^2$


$Trace$ $of$ $ A=$ $a _{1,1}+a _{2,2}+a _{3,3}+...........+a _{n,n}$ 
              $A=$  $2+8+18+.................+2n^2$
              $A=$  $2[1+4+9+................n^2]$

              $A=$  $2 \times [\frac{n(n+1)(2n+1)}{6}]$

              $A=$  $\frac{n(n+1)(2n+1)}{3}$

               $\therefore Opt$ $is$ $[B]$