Question

Using the matrix X = [2,2,2,2,4,6] Calculate M = ( I2 - X(XT X) "-1" XT ) and show that M is an idempotent matrix (Do not convert fractions into decimal numbers). in Assignment Expert

Answer 1

$\text{Given the matrix }X=\begin{bmatrix} 2 & 2 \\ 4 & 6 \end{bmatrix}, \text{we need to calculate }M=[I_2-X(X^TX)^{-1}X^T], \text{where } I_2\\ \text{is the } 2\times 2\text{ identity matrix.}\\ \text{Now,}\\ M=[I_2-X(X^TX)^{-1}X^T]\\ \quad=[I_2-X(X^{-1}(X^T)^{-1})X^T], \text{since }(AB)^{-1}=B^{-1}A^{-1}, \forall \text{ invertible matrix A and B.}\\ \quad=[I_2-X(X^{-1}(X^T)^{-1}X^T)]=[I_2-X(X^{-1})I_2]\\ \quad=[I_2-I_2]=0_2, \text{where }0_2 \text{ is the }2\times 2\text{ zero matrix.}\\ \text{Thus, }\\ M=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\\ \quad\\ \quad\\ \text{Next, to show that M is idempotent we need to show that }M^2=M.\\ M^2=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\times\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=M$