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

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

  1. an Orthogonal Matrix

  2. an Idempotent Matrix

  3. a Nilpotent Matrix

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
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,
$B^{T}=B^{-1}\Rightarrow B\text{ is Orthogonal Matrix}$

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

  1. FFTT

  2. TTFF

  3. FFFT

  4. TTTF
Reveal answer Fill a bubble to check yourself
B Correct answer
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.