Mathematics

Linear Algebra

510 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 $A$ be the $2\times2$ matrices given by $A=\left[a _{ij}\right]$ where $a _{ij} = \left{0,1,2,3,4\right}$ such that $a _{11} + a _{12} + a _{21} + a _{22} = 4$
Find the number of matrices $A$ such that the trace of $A$ is equal to 4

  1. 3

  2. 4

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $tr(A)=4$
$\Rightarrow a _{11}+a _{22}=4$

$a _{ ij }={ { 0,1,2,3,4}  }$

So, diagonal entries of A can be 0 and 4 , 4 and 0, 1 and 3, 3 and 1, 2 and 2,
Hence, 5 matrices are possible

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 $tr(A)+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 $ =

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

The expression is a geometric series of traces. Given the matrices, the sum converges to a specific value based on the properties of the trace and matrix multiplication.

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

Consider 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 the value of the sum 
$ tr(A)+tr \cfrac {(ABC) } {2} +tr \cfrac {A( {BC})^2} {4}+ \cfrac {A( {BC})^3} {2}  +...+ \infty               $is

  1. 6

  2. 9

  3. 12

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

The matrix product BC results in the identity matrix, meaning powers of BC remain the identity matrix. Summing the resulting traces forms a geometric series whose sum evaluates precisely to 6.

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

 $P=\left[ \begin{matrix} { 5a }^{ 2 }+2bc & 6 & 8 \ 13 & { 8b }^{ 2 }-10ac & -9 \ -7 & 5 & { 25c }^{ 2 } \end{matrix} \right]$ and $Q=\left[ \begin{matrix} { a }^{ 2 }+6bc & 3 & 5 \ 12 & { -b }^{ 2 } & 6 \ 1 & 4 & { 17bc }^{ 2 } \end{matrix} \right] a,b$ & $c \epsilon N$, if trace $\left(P\right)=trac\left(Q\right)$, and $a,b$ & $C$ are sides of $\Delta ABC$ with $BC=a,CA=b$ & $AB=C$ then $\cos A$ is:

  1. $\dfrac{-79}{120}$
  2. $\dfrac{-89}{120}$
  3. $\dfrac{-33}{40}$
  4. $\dfrac{-31}{40}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equating the traces of P and Q gives an equation involving a, b, and c. Using the properties of triangle sides and the Law of Cosines, one can solve for cos A.

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

If $\left( \begin{array} { l l } { 3 } & { 2 } \ { 7 } & { 5 } \end{array} \right) A \left( \begin{array} { c c } { - 1 } & { 1 } \ { - 2 } & { 1 } \end{array} \right) = \left( \begin{array} { c c } { 2 } & { - 1 } \ { 0 } & { 4 } \end{array} \right)$  then trace of  $A$  is equal to

  1. $-25$
  2. $-21$
  3. $-15$
  4. $-11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let M1 * A * M2 = M3. Then A = M1^-1 * M3 * M2^-1. Calculate the inverse of the matrices and perform the multiplication to find A, then sum the diagonal elements.

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.$