Tag: properties of matrix multiplication

Questions Related to properties of matrix multiplication

Lets $A=\begin{bmatrix} 0&5 \-5 & 0\end{bmatrix}$ be a skew symmetric matrix and $I + A$ is non singular, then the matrix $B = (I - A)(I + A)^{-1}$ is

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

  1. an Orthogonal Matrix

  2. an Idempotent Matrix

  3. a Nilpotent Matrix

  4. Data Insufficient

Show answer
Correct Option: A
Explanation:

$B=(I-A)(I+A)^{-1}$

$B^{T}=[(I+A)^{-1}]^{T}(I-A)^{T}$
$\Rightarrow B^{T}=[(I+A)^{T}]^{-1}(I-A)^{T}$
$\Rightarrow B^{T}=(I-A)^{-1}(I+A)$             since $A^{T}=-A$
$B^{-1}=(I+A)(I-A)^{-1}$
In this case commutativity holds, so,
$B^{T}=B^{-1}\Rightarrow B\text{ is Orthogonal Matrix}$

Consider the following statements :
$S _1$ : If $f(x)$ and $g(x)$ both are discontinuous function and $f(x) + g(x)$ is continuous, then $f(x) - g(x)$ is discontinuous.

$S _2$ : If a tangent to the standard ellipse $\displaystyle \frac{x^2}{a^2}+\displaystyle \frac{y^2}{b^2} = 1$ intersects the principal axis at A and  B, then least value of $AB$ is $(a+b)$.

$S _3$ : If $A$ and $B$ are two matrices such that $AB = O$, where $O$ is null matrix, then at least one of the matrices $A$ and $B$ must be a null matrix.

$S _4$ : If $a,b,c \epsilon R$ and $D$ is a perfect square of a rational number, then both roots of the quadratic equation $a{ x }^{ 2 }+bx+c=0$ are rational.

State, in order, whether ${ S } _{ 1 },{ S } _{ 2 },{ S } _{ 3 }$ or $ { S } _{ 4 }$ are true or false.
business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

  1. FFTT

  2. TTFF

  3. FFFT

  4. TTTF

Show answer
Correct Option: B
Explanation:

S1 : If $f(x)$ and $g(x)$ both are discontinuous function and $f(x) + g(x)$ is continuous, then $f(x) - g(x)$ is discontinuous.
True
eg: $f(x) = [x]$ and $g(x) = [1-x]$ for non integral values of x, where [.] is a greatest integer function
$f(x)+g(x) = [x]+[1-x] = 0$ is continuous and
$f(x)-g(x)$ is discontinuous.
S2 : If a tangent to the standard ellipse $\displaystyle\frac { x^{ 2 } }{ a^{ 2 } } +\displaystyle\frac { y ^2}{b^2  } =1$  intersects the principal axis at $A$ and  $B$, then least value of $AB$ is $(a+b)$
True
Tanget at $\left( a\cos { \theta , } b\sin { \theta  }  \right)$ is $\displaystyle\frac { x\cos { \theta  }  }{ a } +\displaystyle\frac { y\sin { \theta  }  }{ b } =1$ which intersects axis at $A=(\displaystyle\frac { a }{ \cos { \theta  }  } ,0)$ and $B=(0,\displaystyle\frac { b }{ \sin { \theta  }  })$
$AB^{2} = \displaystyle\frac { a^{ 2 } }{ \cos ^{ 2 }{ \theta  }  } +\displaystyle\frac { b^{ 2 } }{ \sin ^{ 2 }{ \theta  }  } $
AB is minimum at $\theta =\tan ^{ -1 }{ \left(\displaystyle \frac { \sqrt { b }  }{ \sqrt { a }  }  \right)  } $ and minimum value is $a+b$

S3 : If $A$ and $B$ are two matrices such that $AB=O$, where $O$ is null matrix, then at least one of the matrices $A$ and $B$ must be a null matrix.
False
product of two non zero  matrices can be a null matrix.
S4 : If $a,b,c R$ and $D$ is a perfect square of a rational number, then both roots of the quadratic equation $ax^{2}+bx+c=0$ are rational.
False
eg $\sqrt { 3 } x^{ 2 }+\sqrt { 28 } x+\sqrt { 3 } =0$
where $D$ is a perfect square and roots are not rational.