Browse Questions

# Find the product of matrix $A = \begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}$ and its transpose.

Toolbox:
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Given:
$A = \begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}$
Transpose can be obtained by changing the rows and column.
$A' = \begin{bmatrix} 3 & 2 & -5 \end{bmatrix}$
$AA'= \begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}\begin{bmatrix} 3 & 2 & -5 \end{bmatrix}$
$\qquad=\begin{bmatrix} 3(3)+3(2)+3(-5) \\ 2(3)+2(2)+2(-5) \\-5(3)-5(2)-5(-5) \end{bmatrix}$
$\qquad=\begin{bmatrix} 9+6-15 \\ 6+4-10 \\-15-10+25 \end{bmatrix}$
$\qquad=\begin{bmatrix} 0 \\ 0 \\0 \end{bmatrix}$