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

If $A = \begin{bmatrix}1\ 2\ 3
\end{bmatrix}$ then $AA^{1}$.

  1. $40$
  2. $\begin{bmatrix} 1\\ 4\\ 3 \end{bmatrix}$
  3. $\begin{bmatrix} 1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9\end{bmatrix}$

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

If A is a column vector [1, 2, 3]^T, then A^T is the row vector [1, 2, 3]. The product AA^T is a 3x3 matrix: [[1*1, 1*2, 1*3], [2*1, 2*2, 2*3], [3*1, 3*2, 3*3]] = [[1, 2, 3], [2, 4, 6], [3, 6, 9]].

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

Let $A$ be a square matrix such that $A^2 = A$ and $|A| \neq 0$, then choose the correct option.

(A' represents transpose of matrix A)

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

$A$ is a square matrix such that $A^2 = A$ and $|A| \neq 0$
$\Rightarrow A^{-1}A^2=A^{-1}A=I$
$\Rightarrow A=I$
$\therefore A=A'=I$
Hence, option A.

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

For two matrices $A$ and $B$, if $AB=0$, then

  1. $A=0$ and $B=0$
  2. $A=0$ or $B=0$
  3. it is not necessary that $A=0$ or $B=0$

  4. all above are false
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

It is not necessary that, if $AB=0$ then $A=0$ or $B=0$


Take example, $A=\begin{bmatrix}0&0\0&1 \end{bmatrix}, B=\begin{bmatrix}0&1\0&0 \end{bmatrix}$

Clearly $AB=\begin{bmatrix}0&0\0&0 \end{bmatrix}$ but $A,B\neq 0$

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

For any non-singular matrix A, $ \displaystyle A^{-1} $ =

  1. $|A|adj A$
  2. $\dfrac{1}{|A| adj A}$
  3. $\dfrac{adj A}{|A|}$

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

Singular matrix is square matrix whose determinant is equal to Zero.

So, non-singular matrix $A$ is a matrix whose determinant is non-zero.
$\implies$ inverse of $A$ i.e. $A^{-1}$ exist.
$\therefore A^{-1}=\dfrac{adj A}{|A|}$ .... $[|A|\neq 0]$

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

If $A=\begin{bmatrix} \cos { \alpha  }  & -\sin { \alpha  }  \ \sin { \alpha  }  & \cos { \alpha  }  \end{bmatrix}$, $B=\begin{bmatrix} \cos { 2\beta  }  & \sin { 2\beta  }  \ \sin { 2\beta  }  & -\cos { 2\beta  }  \end{bmatrix}$, where 0 < $\beta$ < ${ \pi  }/{ 2 }$, then prove that $BAB=$ ${ A }^{ -1 }$.

  1. True

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

By performing the matrix multiplication BAB and comparing it with the inverse of A (A^-1), one can verify the identity. Given the definitions of A and B, the statement is mathematically true.

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

Let $A$ be a $3\times 2$ matrix with real entries. Let $H = A(A^{T}A)^{-1}A^{T}$ where $A^{T}$ is the transpose of $A$ and let $I$ be the identity matrix of order $3\times 3$. Then

  1. $H^{2} = I$
  2. $H^{2} = -I$
  3. $H^{2} = H$
  4. $H^{2} = -H$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that from the propeties of inverse matrices,

${ \left( A.B \right)  }^{ -1 }={ B }^{ -1 }.A^{ -1 }$
$\Longrightarrow H=A{ A }^{ -1 }{ \left( { A }^{ T } \right)  }^{ -1 }{ A }^{ T }=I\ \Longrightarrow H.H={ H }^{ 2 }=I.H=H$

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

If $ \begin{bmatrix}
              2 & 1 \[0.3em]
              3 & 2
              \end{bmatrix} \ A \begin{bmatrix}
              -3 & 2 \[0.3em]
              5 & -3
              \end{bmatrix} = \begin{bmatrix}
              1 & 0 \[0.3em]
              0 & 1
              \end{bmatrix}$ then the matrix A is equal to

  1. $\begin{bmatrix}

    1 & 1 \\[0.3em]

    1 & 0

    \end{bmatrix}$
  2. $\begin{bmatrix}

    1 & 1 \\[0.3em]

    0 & 1

    \end{bmatrix}$
  3. $\begin{bmatrix}

    1 & 0 \\[0.3em]

    1 & 1

    \end{bmatrix}$
  4. $\begin{bmatrix}

    0 & 1 \\[0.3em]

    1 & 1

    \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

Pre multiplying by inverse of $1^{st}$ matrix on $LHS$
$A \begin{bmatrix} -3 & 2 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix} \, \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}$
$A \begin{bmatrix} -3 & 2 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}$
Post multiplying by $2^{nd}$ matrix.
$A = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix} \begin{bmatrix} -3 & -5 \ -2 & -3 \end{bmatrix}$
$= \begin{bmatrix} 1 &  1 \ 1 & 0 \end{bmatrix}$

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

Find the number of all possible ordered sets of two $(n\times n)$ matrices A and B for which $AB-BA=$$I$.

  1. Infinite
  2. $n^2$
  3. $n!$

  4. Zero
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$AB-BA=I$

Let a $n\times n$ matrix. Take trace of both sides. trace $(AB-BA)=trace \ (I)$

trace $(AB)-$trace $(BA)=n$

$\Rightarrow \ n=0$

so zero possible ordered let of two $(n\times n)$ matrix.
Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A=|a _{ij}| _{2\times 2}$, where $a _{ij}=\left{\begin{matrix} i+j, & if & i\neq j\ i^2-2j, & if & i=j\end{matrix}\right.$, then $A^{-1}=?$

  1. $\dfrac{1}{9}\begin{bmatrix} 0 & 3\\ 3 & 1\end{bmatrix}$
  2. $\dfrac{1}{9}\begin{bmatrix} 0 & -3 \\ -3 & -1\end{bmatrix}$
  3. $\dfrac{1}{9}\begin{bmatrix} 4 & 1\\ -1 & 2\end{bmatrix}$
  4. $\dfrac{-1}{9}\begin{bmatrix} 4 & 1\\ -1 & 2\end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\begin{array}{l} A={ \left| { aij } \right| _{ 2\times 2 } } \ { a _{ 11 } }=1-2=-1 \ { a _{ 12 } }=3 \ { a _{ 21 } }=3 \ { a _{ 22 } }=4-2\times 2=0 \ A=\left[ \begin{array}{l} -1 & 3 \ 3 & D \end{array} \right]  \ { A^{ -1 } }=\frac { 1 }{ { \left| A \right|  } } adjA \ =\frac { { -1 } }{ { 09 } } \left[ \begin{array}{l} 0 & -3 \ -3 & -1 \end{array} \right]  \ =\frac { 1 }{ 9 } \left[ \begin{array}{l} 0 & 3 \ 3 & 1 \end{array} \right]  \end{array}$

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

If A =$\left[ \begin{matrix} i \ 0 \end{matrix}\begin{matrix} 0 \ -1 \end{matrix} \right] $, than check whether: ${{\text{A}}^2} =  - {\text{I,(}}{{\text{i}}^2} =  - 1)$

  1. True

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

Given $A=\begin{bmatrix} i & 0  \ 0 & -1 \end{bmatrix}$ where $i^{2}=-1$


Taking LHS

$\Rightarrow A^{2}=A.A$

$\begin{bmatrix} i & 0 \ 0 & -1 \end{bmatrix}\begin{bmatrix} i & 0 \ 0 & -1 \end{bmatrix}$ (using matrix multiplication)

$\Rightarrow \begin{bmatrix} { i }^{ 2 }+0 & 0+0 \ 0+0 & 0+1 \end{bmatrix}$

$\Rightarrow \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}$ (wihch is not equal to $-I$)

as $I=\begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}$

$\therefore A^{2}=-I$ is not a valid relation.

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

If A and B are matrices of the same order, then $\displaystyle :\left ( A+B \right )^{2}= A^{2}+2AB+B^{2}$ is possible, iff

  1. AB= I

  2. BA= I

  3. AB= BA

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

$\displaystyle :\left ( A+B \right )^{2}=(A+B)(A+B)$
$= A^{2}+AB+BA+B^{2}$
If $AB=BA$ then 
$\left ( A+B \right )^{2}=A^{2}+2AB+B^{2}$


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.