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

If $\omega$ is the complex cube root of unity, then inverse of $\begin{bmatrix} \omega  & 0 & 0 \ 0 & { \omega  }^{ 2 } & 0 \ 0 & 0 & { \omega  }^{ 2 } \end{bmatrix}$ is

  1. $\begin{bmatrix} -\omega & 0 & 0 \\ 0 & { \omega }& 0 \\ 0 & 0 & { \omega }^{ 2 } \end{bmatrix}$
  2. $\begin{bmatrix} \omega^{2} & 0 & 0 \\ 0 & { \omega }& 0 \\ 0 & 0 & 1 \end{bmatrix}$
  3. $\begin{bmatrix} \omega^{3} & 0 & 0 \\ 0 & { \omega } & 0 \\ 0 & 0 & 1 \end{bmatrix}$
  4. $\begin{bmatrix} \omega & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & { \omega }^{ 2 } \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$M=\begin{bmatrix} w & 0 & 0 \\ 0 & { w }^{ 2 } & 0 \\ 0 & 0 & { w }^{ 2 } \end{bmatrix}$
$(M)=w3w^2.w^3$
$=w^3.w^2$
$=w^3(w^3=1)$
$=(1)$

$adjA\Rightarrow \begin{bmatrix} { w }^{ 5 } & 0 & 0 \\ 0 & { w }^{ 4 } & 0 \\ 0 & 0 & { w }^{ 3 } \end{bmatrix}$

${ A }^{ -1 }=\dfrac { 1 }{ 1 } \begin{bmatrix} { w }^{ 2 }.{ w }^{ 3 } & 0 & 0 \\ 0 & { w }.{ w }^{ 3 } & 0 \\ 0 & 0 & \left( { w }^{ 3 } \right)  \end{bmatrix}$

$=\dfrac { 1 }{ 1 } \begin{bmatrix} { w }^{ 2 }.{ w }^{ 3 } & 0 & 0 \\ 0 & { w }.{ w }^{ 3 } & 0 \\ 0 & 0 & \left( { w }^{ 3 } \right)  \end{bmatrix}$
option $B$ is correct

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

The inverse of the matrix $\begin{bmatrix}1 & 0 & 1\ 0 & 2 & 3\ 1 & 2& 1\end{bmatrix}$ is

  1. $\dfrac {-1}{6} \begin{bmatrix}-4 & 2 & -2\\ 3 & 0 & -3\\ -2 & -2& 2\end{bmatrix}$
  2. $\dfrac {1}{6} \begin{bmatrix}-4 & 2 & -2\\ 3 & 0 & -3\\ -2 & -2& 2\end{bmatrix}$
  3. $\begin{bmatrix}-2 & 1 & -1\\ 1 & 0 & -1\\ -2 & -2& 2\end{bmatrix}$
  4. $\begin{bmatrix}2 & -1 & 1\\ -1 & 0 & 1\\ 2 & 2& -2\end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse of a matrix A is (1/det(A)) * adj(A). For the given matrix, the determinant is 1(2-6) - 0 + 1(0-2) = -4 - 2 = -6. Calculating the adjugate matrix and dividing by -6 yields the correct inverse.

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 $M = \left[ \begin{array}{l}0\,\,\,\,2\5\,\,\,\,\,0\end{array} \right]\,\,\,and\,\,N = \left[ \begin{array}{l}0\,\,\,\,5\2\,\,\,\,\,0\end{array} \right]$,then ${M^{2011}}$ is-

  1. ${10^{1005}}M$
  2. ${10^{1005}}N$
  3. ${10^{2010}}M$
  4. ${10^{2011}}M$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Computing M^2 gives matrix multiplication resulting in [10 0; 0 10], which is 10I. By observing patterns for higher powers, M^(2k) = 10^k I and M^(2k+1) = 10^k M. For M^2011, since 2011 is odd, it simplifies to 10^1005 M.

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

If $A = \left[ \begin{array}{l}\cos \theta \,\,\,\,\sin \theta \ - \sin \theta \,\,\,\cos \theta \end{array} \right]$ where $\theta  = \frac{{2\pi }}{{19}}$ then ${A^{2017}} = $

  1. $A$
  2. ${A^3}$
  3. ${A^5}$
  4. $i$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A is a rotation matrix R(theta). A^n = R(n*theta). Here, A^2017 = R(2017 * 2pi / 19). Since 2017 = 19 * 106 + 3, A^2017 = R(3 * 2pi / 19) = A^3.

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.