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 business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

The matrices $\begin{bmatrix} \cos { \theta  }  & -\sin { \theta  }  \ \sin { \theta  }  & \cos { \theta  }  \end{bmatrix}$ and $\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}$ commute under multiplication

  1. if $a=b$ or $\theta=n\pi,$ where $n$ is an integer

  2. always

  3. never
  4. if $a\cos { \theta } \neq b\sin { \theta } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\begin{bmatrix} \cos { \theta  }  & -\sin { \theta  }  \ \sin { \theta  }  & \cos { \theta  }  \end{bmatrix}\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}=\begin{bmatrix} a\cos { \theta  }  & -b\sin { \theta  }  \ a\sin { \theta  }  & b\cos { \theta  }  \end{bmatrix}$   ...(1)

And $\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}\begin{bmatrix} \cos { \theta  }  & -\sin { \theta  }  \ \sin { \theta  }  & \cos { \theta  }  \end{bmatrix}=\begin{bmatrix} a\cos { \theta  }  & -a\sin { \theta  }  \ b\sin { \theta  }  & b\cos { \theta  }  \end{bmatrix}$   ...(2)
From (1) and (2), we get
$a\sin { \theta  } =b\sin { \theta  } \Rightarrow \left( a-b \right) \sin { \theta  } =0$
either $a=b$ or $\sin { \theta  } =0$
$\Rightarrow\theta=n\pi;n\in Z$

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A$ and $B$ are two square matrices of order $3 \times  3$ which satisfy $AB = A$ and $BA = B$, then Which of the following is true?

  1. If matrix $A$ is singular, then matrix $B$ is non singular.
  2. If matrix $A$ is nonsingular, then matrix $B$ is singular.
  3. If matrix $A$ is singular, then matrix $B$ is also singular.

  4. Cannot say anything.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$A$ and $B$ are two square matrices of order $3\times3$ which satisfy $AB=A$ and $BA=B,$ then if matrix $A$ is singular, then matrix $B$ is also singular .
$A$ singular matrix is that matrix whose determinants is zero and which is non-irreversible.
$\therefore A$ and $B$ both are singular matrices, then only the conditions $AB=A$ and $BA=B$ holds true.
Hence, the answer is if matrix $A$ is singular, then matrix $B$ is also singular.
Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

 In matrices $AB = O$ does not necessarily mean that 

  1. $A=0$
  2. $B=0$
  3. Both $ A = 0$ and $B=0$

  4. all of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $\quad A = \begin{bmatrix}1 & -1 & 1 \ -3 & 2 & -1

\ -2 & 1 & 0\end{bmatrix}$ and $B = \begin{bmatrix}1&2

& 3\2&4&6 \ 1&2 &3\end{bmatrix}$

$\therefore\quad

AB = \begin{bmatrix}1 & -1 & 1 \ -3 & 2 & -1 \ -2

& 1 & 0\end{bmatrix}\times\begin{bmatrix}1&2 &

3\2&4&6 \ 1&2 &3\end{bmatrix}$

$\quad                   = \begin{bmatrix}0&0&0 \ 0&0&0 \ 0&0&0\end{bmatrix} = O$

$\therefore \quad AB = O$
But neither $A = O$ nor $B = O$.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If matrix $A = [a _{ij}] _{2\times 2}$, where $a _{ij} = \left{\begin{matrix} 1,& \ \text{if}\ &i\neq j \ 0, & \ \text{if}\ & i + j\end{matrix}\right.$, then $A^{2}$ is equal to

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

$A=\begin{bmatrix}  a _{11}& a _{12} \  a _{21}&  a _{22}\end{bmatrix}$


$a _{11}:1\,\,\,a _{12}:0$

$a _{21}:0\,\,\,a _{22}:1$

$A=\begin{bmatrix}  1&  0\ 0 &  1\end{bmatrix}$

$A^2=\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \ 0 &  0\end{bmatrix}=\begin{bmatrix}  1& 0 \  0&  1\end{bmatrix}=I$

$\boxed{Hence\,A^2=I}$

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A = \begin{bmatrix} -2& 3\ 1 & 1\end{bmatrix}$ then $|A^{-1}| = ?$

  1. $-5$
  2. $\dfrac {-1}{5}$
  3. $\dfrac {1}{25}$
  4. $25$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$AA^{-1} = I\Rightarrow |AA^{-1}| = |I|\Rightarrow |A|\cdot |A^{-1}| = 1\Rightarrow |A^{-1}| = \dfrac {1}{|A|}$.
$|A| = \begin{vmatrix}-2 & 3\ 1 & 1\end{vmatrix} = (-2 - 3) = (-5) \Rightarrow |A^{-1}| = \dfrac {-1}{5}$.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Let $A$ and $B$ be two $2 \times 2$ matrices. Consider the statements
          $(i)$ $AB =0 \Rightarrow A = 0 :or :B = 0$
         $ (ii)$ $AB =I \Rightarrow A =B^{-1}$
          $(iii)$ $(A + B)^2 = A^2 + 2AB + B^2$

  1. $(i)$ is false, $(ii)$ and $(iii)$ are true
  2. $(i)$ and $(iii)$ are false, $(ii)$ is true
  3. $(i)$ and $(ii)$ are false, $(iii)$ is true
  4. $(ii)$ and $(iii)$ are false, $(i)$ is true
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(i) is false.


If $A=\begin{bmatrix}0& 1\0 & -1\end{bmatrix}$ and $B =\begin{bmatrix}1& 1\0 & 0\end{bmatrix}$, then

 $AB=\begin{bmatrix}0& 0\0 & 0\end{bmatrix}=0$

Thus,  $AB = 0$ but neither $A = 0$ nor $B = 0$

(iii) is false since matrix multiplication is not commutative.

$(A + B)^2 = A^2 + AB + BA + B^2$

(ii) is true as the product $AB$ is an identity matrix, if and only $B$ is inverse of the matrix $A$.

Hence, option B.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A = \begin{bmatrix} 2& -1\ 1 & 3\end{bmatrix}$, then $A^{-1} = ?$

  1. $\begin{bmatrix}\dfrac {3}{7}
    &\dfrac {-1}{7} \\
    \dfrac {1}{7} & \dfrac {2}{7}
    \end{bmatrix}$
  2. $\begin{bmatrix}\dfrac {3}{7}
    &\dfrac {1}{7} \\
    \dfrac {-1}{7} & \dfrac {2}{7}
    \end{bmatrix}$
  3. $\begin{bmatrix}\dfrac {3}{7}
    &\dfrac {1}{7} \\
    \dfrac {1}{7} & \dfrac {2}{7}
    \end{bmatrix}$

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

$|A| = \begin{vmatrix}2 & -1\ 1 & 3\end{vmatrix} = (6 + 1) = 7\neq 0$
$M _{11} = 3, M _{12} = 1, M _{21} = -1$ and $M _{22} = 2$


$\therefore C _{11} = 3, C _{12} = -1, C _{21} = 1$ and $C _{22} = 2$

$\Rightarrow Adj\ A = \begin{bmatrix} 3& -1\ 1 & 2\end{bmatrix}^T = \begin{bmatrix} 3& 1\ -1 & 2\end{bmatrix}$

$\Rightarrow A^{-1} = \dfrac {1}{|A|} (adj\ A) = \dfrac {1}{7} \begin{bmatrix}3 & 1\ -1 & 2\end{bmatrix} = \begin{bmatrix}\dfrac {3}{7} & \dfrac {1}{7}\ -\dfrac {1}{7} & \dfrac {2}{7}\end{bmatrix}$.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A$ and $B$ are two square matrices of the same order and $m$ is a positive integer, then
$(A + B)^m =$ $^mC _0A^m +$ $^mC _1 A^{m -1} B + ^mC _2A^{m-2} B^2 + ... +$ $^mC _{m- 1} AB^{m-1}+$ $^mC _m B^m$ if

  1. $AB =BA$
  2. $AB + BA =0$
  3. $A^m = 0, \:B^m = 0$

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

Binomial theorem is applicable if and only if $AB=BA$.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Let $A, : B : and : C$ be $2\times 2$ matrices with entries from the set of real numbers. Define $\ast $ as follows: $\displaystyle A\ast B=\frac{1}{2}(AB + BA)$, then

  1. $A\ast B=B\ast A$
  2. $A\ast A=A^2$
  3. $A\ast (B+C)=A\ast B+A\ast C$
  4. $A\ast I=A$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

$\displaystyle A*1=\frac { 1 }{ 2 } \left( AI+IA \right) =\frac { 1 }{ 2 } \left( A+A \right) =A$

$\therefore A*I=A,\therefore$ (a)holds.
$\displaystyle A*A=\frac { 1 }{ 2 } \left( AA+AA \right) =\frac { 1 }{ 2 } \left( { A }^{ 2 }+{ A }^{ 2 } \right) ={ A }^{ 2 }$
$\therefore$ (b) holds.
$\displaystyle \left( c \right) A*B=\frac { 1 }{ 2 } \left( AB+BA \right) B*A=\frac { 1 }{ 2 } \left( BA+AB \right) =\frac { 1 }{ 2 } \left( AB+BA \right) $
$[\because $ addition is commutative$]$
$\therefore A*B=B*A,\therefore$ (c)holds
$\displaystyle A*\left( B+C \right) =\frac { 1 }{ 2 } \left( A\left( B+C \right) +\left( B+C \right) A \right) =\frac { 1 }{ 2 } \left( AB+CA+BA+CA \right)$
$\displaystyle  =\frac { 1 }{ 2 } \left( AB+BA \right) +\frac { 1 }{ 2 } \left( AC+CA \right) =A*B+A*C$
$\therefore$ (d) holds.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A^k=0$ for some value of $k$ and $B=1+A+A^2+...+A^{k-1},$ then $B^{-1}$ equal

  1. $I-A$
  2. $I+A$
  3. $I-A^{k-1}$

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

Let $\displaystyle B=I+A+{ A }^{ 2 }+{ A }^{ 3 }+...+{ A }^{ k-1 }$

$\displaystyle \therefore B\left( I-A \right) =\left( I+A+{ A }^{ 2 }+...+{ A }^{ k-1 } \right) \left( I-A \right) $
$\displaystyle =I-A+A-{ A }^{ 2 }+{ A }^{ 2 }+...+{ A }^{ k-1 }-{ A }^{ k }=I={ A }^{ k }=I\quad \quad \quad \left( \because { A }^{ k }=O \right) $
Hence, $\displaystyle { \left( I-A \right)  }^{ -1 }=I+A+{ A }^{ 2 }+...+{ A }^{ k }-1=B$
$\displaystyle \Rightarrow { B }^{ -1 }=I-A$

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Let $A, : B : and : C$ be $2\times 2$ matrices with entries from the set of real numbers. Define $\ast $ as follows:
  $\displaystyle A \ast B=\frac{1}{2}(AB\,'+A'B)$. Which of the given is true?

  1. $A\ast B= B \ast A$
  2. $A\ast A=A^2$
  3. $A\ast (B+C)=A\ast B+A \ast C$
  4. $A\ast I =A+A'$
Reveal answer Fill a bubble to check yourself
A,C Correct answer
Explanation

$\displaystyle A \ast B=\frac{1}{2}(AB'+A'B)$

1) $\displaystyle B \ast A=\frac{1}{2}(BA'+B'A)=\frac{1}{2}(AB'+A'B)=A \ast B$

2)$\displaystyle A \ast A=\frac{1}{2}(AA'+A'A)$

3)$\displaystyle A \ast (B+C)=\frac{1}{2}(A(B+C)'+A'(B+C))$

                             $=\displaystyle\frac{1}{2}(AB'+A'B)+\frac{1}{2}(AC'+A'C)$

                             $=A\ast B+A \ast C$

4)$\displaystyle A \ast I=\frac{1}{2}(AI'+A'I)=\frac{1}{2}(A+A')$

Hence, options A and C.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Say true or false:

Let A, B be two matrices such that they commute, then $(AB)^n = A^nB^n$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A$ and $B$ commute each other then $AB=BA$
${ \left( AB \right)  }^{ n }={ \left( BA \right)  }^{ n }\ \Rightarrow { \left( AB \right)  }^{ n }={ B }^{ n }{ A }^{ n }={ A }^{ n }{ B }^{ n }={ \left( BA \right)  }^{ n }$