Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

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

Let three matrices $A=\begin{bmatrix} 2 & 1\ 4 & 1\end{bmatrix}; B\begin{bmatrix} 3 & 4\ 2 & 3\end{bmatrix}$ and $C=\begin{bmatrix} 3 & -4\ -2 & 3\end{bmatrix}$ then $t _r(A)+t _r\left(\dfrac{ABC}{2}\right)+t _r\left(\dfrac{A(BC)^2}{4}\right)+t _r\left(\dfrac{A(BC)^3}{8}\right)+.....+\infty =?$

  1. $6$
  2. $9$
  3. $12$

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

This is identical to the previous series summation problems involving matrix traces.

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

If $A=\begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix}$, $B=\begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix}$ and $C=\begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix}$, then $\displaystyle tr(A)+tr\left(\frac{ABC}{2}  \right)+tr\left(\frac{A{(BC)}^{2}}{4}  \right)+tr\left(\frac{A{(BC)}^{2}}{8}  \right)+...+\infty=  $

  1. $6$
  2. $9$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This has a typo in the series - the 4th term should be A(BC)³/8, but it repeats A(BC)²/8. Assuming the intended pattern (powers of BC increase): BC = I, tr(A) = 3. Series = 3[1 + 1/2 + 1/4 + 1/8 + ...] = 3 x 2 = 6. Answer A is correct despite the typographical error.

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

Let $\displaystyle A=\begin{bmatrix}-1\2\3\end{bmatrix}$ and $\displaystyle B=\begin{bmatrix} -2 & -1 & -4 \end{bmatrix}$

If trace of matrix $AB$ is $-12$, then the value of $k$ 

  1. $7$
  2. $1$
  3. $2$

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

$\displaystyle A=\begin{bmatrix}-k\2\3\end{bmatrix}$ 


$\displaystyle B=\begin{bmatrix}-2-1-4\end{bmatrix}$

$ \therefore AB=\begin{bmatrix}2k &k  &4k \-4  &-2  &-8 \-6  &-3  &-12 \end{bmatrix}$ 

The trace (often abbreviated to tr) of a square matrix A is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A.

$\displaystyle \therefore Tr(AB)=$Summation of diagonal elements$=12$

                    $\Rightarrow +2k-2-12=-12$...............(According to question)

                    $\Rightarrow 2k=14-12=2$

                    $\Rightarrow k=1$

                    $\therefore k=1$ 

                    Hence, option B.

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

Let $A$ and $B$ are two matrices of same order $\displaystyle 3\times 3$ given by $\displaystyle A=\begin{bmatrix}1 &3  &\lambda+2 \\2  &4  &6 \\3  &5  &8 \end{bmatrix}$ $\displaystyle B= \begin{bmatrix}3 &2  &4 \\3  &2  &5 \\2
 &1  &4 \end{bmatrix}$If $\displaystyle \lambda =4$,then $\displaystyle \frac {1}{6}\left \{tr(AB)+tr(BA)  \right \} $ is equal to
  1. $42$
  2. $37$
  3. $35$

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

$A=\begin{bmatrix} 1 & 3 & 6 \ 2 & 4 & 6 \ 3 & 5 & 8 \end{bmatrix},B=\begin{bmatrix} 3 & 2 & 4 \ 3 & 2 & 5 \ 2 & 1 & 4 \end{bmatrix}$

$ AB=\begin{bmatrix} 1 & 3 & 6 \ 2 & 4 & 6 \ 3 & 5 & 8 \end{bmatrix}\begin{bmatrix} 3 & 2 & 4 \ 3 & 2 & 5 \ 2 & 1 & 4 \end{bmatrix}=\begin{bmatrix} 24 & 14 & 43 \ 30 & 18 & 52 \ 40 & 24 & 69 \end{bmatrix}$


$\Rightarrow  tr(AB)=24+18+69=111$

$tr(BA)=tr(AB)=111$

 $\displaystyle \frac {1}{6}\left {tr(AB)+tr(BA)  \right }=\frac{1}{6}222 =37$

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

Let $A$ and $B$ are two matrices of same order $\displaystyle 3\times 3$ given by 
$\displaystyle A=\begin{bmatrix}1 &3  &\lambda+2 \2  &4  &6 \3  &5  &8 \end{bmatrix},$ $\displaystyle B= \begin{bmatrix}3 &2  &4 \3  &2  &5 \2 &1  &4 \end{bmatrix}$

If $A$ is a singular matrix, then $tr(A + B)$ is equal to

  1. $24$
  2. $11$
  3. $22$

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

Given, $A=\begin{bmatrix} 1 & 3 & \lambda +2 \ 2 & 4 & 6 \ 3 & 5 & 8 \end{bmatrix},B=\begin{bmatrix} 3 & 2 & 4 \ 3 & 2 & 5 \ 2 & 1 & 4 \end{bmatrix}$

$A+B=\begin{bmatrix} 4 & 5 & \lambda +6 \ 5 & 6 & 11 \ 5 & 6 & 12 \end{bmatrix}$

$tr(A+B)=4+6+12=22$

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

The trace of the matrix $A = \begin{bmatrix}1 & -5 & 7\ 0 & 7 & 9\ 11 & 8 & 9\end{bmatrix}$ is

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

Given,  $A = \begin{bmatrix}1 & -5 & 7\ 0 & 7 & 9\ 11 & 8 & 9\end{bmatrix}$ .


Now trace of $A=$ sum of the diagonal elements of $A$.

So trace of $A=1+7+9=17$.

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

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

  1. $nk$
  2. $n + k$
  3. $\dfrac {n}{k}$

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

Given $A = [a _{ij}]$ is a scalar matrix of order $n\times n$ such that $a _{ii} = k$ for all $i$.


The trace of a square matrix is defined to be the sum of the diagonal elements.

Now, trace $(A)=\displaystyle\sum\limits _{i=1}^n a _{ii}=$$\displaystyle\sum\limits _{i=1}^n k=nk.$

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

If $A$ is a $3\times 3$ skew-symmetric matrix, then trace of $A$ is equal to

  1. $1$
  2. $|A|$
  3. $-1$

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

As $A$ is a skew symmetric matrix
                   $A' = -A$
$\Rightarrow  a _{ii}=0 : \forall : i \Rightarrow trace : (A) =0$
Also             $|A|=|A'|=|-A|=(-1)^3|A|$
$\Rightarrow  2|A|=0\Rightarrow |A|=0$
Hence, option B.

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

If $A$ is $2\times 2$ matrix such that $A^2 = 0$, then $tr :(A)$ is

  1. 1

  2. 0

  3. -1

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

If $A = 0,tr(A)=0$.

Suppose $A\neq 0$ and $A =\begin{bmatrix}a

&b \c &d \end{bmatrix}$, then $|A|=0$ and


$A^2-(a+d)A+ad-bc=0$

$\Rightarrow a+d=0$

$\therefore tr(A)=0$

Hence, option B.

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

For $\alpha, \beta, \gamma \in R$, let $A=\begin{bmatrix} { \alpha  }^{ 2 } & 6 & 8 \ 3 & { \beta  }^{ 2 } & 9 \ 4 & 5 & { \gamma  }^{ 2 } \end{bmatrix}$ and $B=\begin{bmatrix} 2\alpha  & 3 & 5 \ 2 & 2\beta  & 6 \ 1 & 4 & 2\gamma -3 \end{bmatrix}$. If ${ T } _{ r }(A)={ T } _{ r }(B)$ then the value of $\left( \cfrac { 1 }{ \alpha  } +\cfrac { 1 }{ \beta  } +\cfrac { 1 }{ \gamma  }  \right) $ is-

${ T } _{ r }(A)$ is a Trace(A) of a matrix

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ T } _{ r }(A)={ \alpha  }^{ 2 }+{ \beta  }^{ 2 }+{ \gamma  }^{ 2 }$


${ T } _{ r }(B)=2\alpha +2\beta +2\gamma -3$

$\Rightarrow { \alpha  }^{ 2 }+{ \beta  }^{ 2 }+{ \gamma  }^{ 2 }=2\alpha +2\beta +2\gamma -3$

$\Rightarrow

{ \left( \alpha -1 \right)  }^{ 2 }+{ \left( \beta -1 \right)  }^{ 2

}+{ \left( \gamma -1 \right)  }^{ 2 }=0\quad $
$\Rightarrow \quad

\alpha =\beta =\gamma =1\Rightarrow \quad \cfrac { 1 }{ \alpha  }

+\cfrac { 1 }{ \beta  } +\cfrac { 1 }{ \gamma  } =3$

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

i. Trace of the matrix is called sum of the elements in a principle diagonal of the square matrix. 
ii. The trace of the matrix $\begin{bmatrix}
8 & 7 &5\
5 &8 & 2\
7 & 2 & 8
\end{bmatrix}$ is 24 Which of the following statement is correct. 

  1. Only i

  2. Only ii

  3. Both i and ii

  4. Neither i nor ii
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$(i)\rightarrow$definition of matrix
$(ii)$ Sum of elements in principle diagonal
$=8+8+8=24$
Option is correct
Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $A=\begin{bmatrix}
1 &4  &7 \
2 &6  &5 \
3 &-1  &2
\end{bmatrix}$ and B $=$ diag (1 2 5), then
trace of matrix $AB^{2}$ is

  1. 74

  2. 75

  3. 529

  4. 23
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $B=diag(1    2    5)$
$\Rightarrow B^{2}=diag(1     4    25)$

Now, $AB^{2}=\begin{bmatrix}1 &0  &0 \ 0 &24  &0 \ 0 &0  &50 \end{bmatrix}$

$\Rightarrow tr(AB^{2})=1+24+50=75$

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

Let three matrices $A=\begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix}$; $B=\begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix}$ and $C=\begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix}$ then find
${ tr }\left( A \right) +{ tr }\left( \dfrac { ABC }{ 2 }  \right) { tr }\left( \dfrac { A{ \left( BC \right)  }^{ 2 } }{ 4 }  \right) +{ tr }\left( \dfrac { A{ \left( BC \right)  }^{ 3 } }{ 8 }  \right) +....+\infty $, where $tr(A)$ represents trace of matrix $A$.

  1. $6$
  2. $9$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A=\left| \begin{matrix} 2 & 1 \ 4 & 1 \end{matrix} \right| ,\quad B=\left| \begin{matrix} 3 & 4 \ 2 & 3 \end{matrix} \right| ,\quad C=\left| \begin{matrix} 3 & -4 \ -2 & 3 \end{matrix} \right| \ AB\quad =\quad \left| \begin{matrix} 2 & 1 \ 4 & 1 \end{matrix} \right| \left| \begin{matrix} 3 & 4 \ 2 & 3 \end{matrix} \right| \ \qquad =\quad \left| \begin{matrix} 6+2 & 8+3 \ 12+2 & 16+3 \end{matrix} \right| \ \quad \quad \quad =\quad \left| \begin{matrix} 8 & 11 \ 14 & 19 \end{matrix} \right| \ ABC\quad =\quad \left| \begin{matrix} 8 & 11 \ 14 & 19 \end{matrix} \right| \left| \begin{matrix} 3 & -4 \ -2 & 3 \end{matrix} \right| \ \quad \quad \quad \quad \quad =\quad \left| \begin{matrix} 24-22 & -32+33 \ 42-38 & -56+57 \end{matrix} \right| \ \qquad \quad \quad =\quad \left| \begin{matrix} 2 & 1 \ 4 & 1 \end{matrix} \right| \ { (BC) }^{ 2 }\quad =\quad \left| \begin{matrix} 3 & 4 \ 2 & 3 \end{matrix} \right| \left| \begin{matrix} 3 & -4 \ -2 & 3 \end{matrix} \right| \ \qquad \quad \quad =\quad { \left| \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} \right|  }^{ 2 }$

$tr(A)\quad +\quad tr(\frac { ABC }{ 2 } )\quad +\quad tr(\frac { A{ (BC) }^{ 2 } }{ 4 } )\quad +\quad tr(\frac { A{ (BC) }^{ 3 } }{ 8 } )\quad +\quad .\quad .\quad .\quad +\quad \infty $
$\quad =\quad 3\quad +\quad 2\quad +\quad 1\quad +\quad 0\quad +\quad 0\quad +\quad 0\quad +\quad .\quad .\quad .\quad +\quad \infty \ \quad =\quad 6$

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

Elements of a matrix $A$ of order $10\times10$ are defined as ${ a } _{ ij }={ w }^{ i+j }$(where $w$ is cube root of unity), then trace ($A$) of the matrix is

  1. $0$
  2. $1$
  3. $3$

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

Here, $A=\begin{bmatrix} { \omega  }^{ 2 } & { \omega  }^{ 3 } & ... & { \omega  }^{ 11 } \ { \omega  }^{ 3 } & { \omega  }^{ 4 } & ... & { \omega  }^{ 12 } \ ... & ... & ... & ... \ { \omega  }^{ 11 } & { \omega  }^{ 12 } &  & { \omega  }^{ 20 } \end{bmatrix}$

$tr(A)={ \omega  }^{ 2 }+{ \omega  }^{ 4 }+{ \omega  }^{ 6 }+.....+{ \omega  }^{ 20 }$

$=\displaystyle \frac { { \omega  }^{ 2 }(1-{ \omega  }^{ 20 }) }{ 1-{ \omega  }^{ 2 } } $