# Calculating the standard matrix for orthogonal projection, 3 way matrix multiplication

I'm trying to calculate the standard matrix for orthogonal projection using the formula for the matrix P=A(A^T A)^-1 A^T

My approach has diverged from the supplied answer to this question

It seems that I've done the working for the inner part (A^T A) correctly but when it comes to multiplying the outer parts I have a very different answer. I've produced a 1x1 matrix while the supplied answer has a 3x3 matrix.

I really just started at the left and multiplied the outer A by the inverse of the inner multiplication: 1/21, then multiplied the resulting vector with the outer A^T .

For some reason I can't get Wolfram Alpha to calculate the matrix product either, it stops after the 1st multiplication Wolfram Alpha Equation

Would anyone be able to let me know what my mistake has been, and how I can properly approach this type of question in the future?

You make a mistake when multiplying the column vector with the row vector. Remember that $$\begin{bmatrix} a \\ b \end{bmatrix} \begin{bmatrix} c & d \end{bmatrix} = \begin{bmatrix} ac & ad \\ bc & bd \end{bmatrix}$$