If X is a 2- 3 matrix such that - XT X - 0 and A-I2-XXTX-1XT then A2 is equal to- XT denotes transpose of matrix X

The correct option is B A By property of inverse, (AB)^ 1=B^ 1A^ 1 So, A=I_2 X(X^TX)^ 1X^T ⇒ nbsp;A=I_2 XX^ 1(X^T)^ 1X^T A=I_2 (I_2)(I ...

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Byju's Answer

Standard XII

Mathematics

Inverse of a Function

If X is a ...

Question

If $X$ is a $2 \times 3$ matrix such that $∣ ∣ X^{T} X ∣ ∣ \neq 0$ and $A = I_{2} - X (X^{T} X)^{- 1} X^{T}$ then $A^{2}$ is equal to:
( $X^{T}$ denotes transpose of matrix $X$ )

A

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I

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A^{- 1}

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A X

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Solution

The correct option is B $A$
By property of inverse,
$(A B)^{- 1} = B^{- 1} A^{- 1}$
So, $A = I_{2} - X (X^{T} X)^{- 1} X^{T}$
$\Rightarrow A = I_{2} - X X^{- 1} (X^{T})^{- 1} X^{T}$
$A = I_{2} - (I_{2}) (I_{2})$
(because $M M^{- 1} = I = M^{- 1} M a n d I_{2} = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$ )
So, $A = I_{2} - I_{2}^{2} = [\begin{matrix} 0 & 0 0 & 0 \end{matrix}]$
So, $A^{2} = A$

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If X is a 2×3 matrix such that ∣∣XTX∣∣≠0 and A=I2−X(XTX)−1XT then A2 is equal to:(XT denotes transpose of matrix X)

The correct option is B ABy property of inverse,(AB)−1=B−1A−1So,A=I2−X(XTX)−1XT⇒A=I2−XX−1(XT)−1XTA=I2−(I2)(I2)(because MM−1=I=M−1MandI2=[1001])So,A=I2−I22=[0000]So,A2=A

If $X$ is a $2 \times 3$ matrix such that $∣ ∣ X^{T} X ∣ ∣ \neq 0$ and $A = I_{2} - X (X^{T} X)^{- 1} X^{T}$ then $A^{2}$ is equal to:
( $X^{T}$ denotes transpose of matrix $X$ )