Tag: properties of multiplication of matrix

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

If $A=\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix}$ then $A^n=\begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \ 3^{n-1} & 3^{n-1} & 3^{n-1} \ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}$ , $n \in N$

  1. True

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

For A = [[1,1,1],[1,1,1],[1,1,1]], A^2 = 3A, A^3 = 9A = 3^2 * A. By induction, A^n = 3^(n-1) * A. This matches the given matrix.

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

If f (x)= $\left[ {\begin{array}{{20}{c}}  {\cos \,x}&{ - \sin \,x}&0 \   {\sin \,x}&{\cos \,x\,}&0 \   0&0&1 \end{array}} \right]$ and $\left[ {\begin{array}{{20}{c}}  {\cos \,x}&0&{\sin \,x} \   0&1&0 \   { - \sin \,x}&0&{\cos \,x} \end{array}} \right]$
then ${\left[ {f\left( x \right)g\left( y \right)} \right]^{ - 1}}$ is equal to 

  1. $f\left( { - x} \right)g\left( { - y} \right)$
  2. $f\left( {{x^{ - 1}}} \right)g\left( {{y^{ - 1}}} \right)$
  3. $g\left( { - y} \right)f\left( { - x} \right)$
  4. $g\left( {{y^{ - 1}}} \right)f\left( {{x^{ - 1}}} \right)$
Reveal answer Fill a bubble to check yourself
B Correct answer
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,

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