Question-7

\(A\) is a square matrix whose columns are \(C_{1}\) and \(C_{2}\) and \(B=\begin{bmatrix}b_{11} & b_{12}\\ b_{21} & b_{22} \end{bmatrix}\). What are the first and second columns of \(AB\)?

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We have \(A=\begin{bmatrix}\vert & \vert\\ C_{1} & C_{2}\\ \vert & \vert \end{bmatrix}\). Let us try to understand what happens when we multiply \(A\) with a vector \(x=\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix}\). We have:

\[ Ax=x_{1}C_{1}+x_{2}C_{2} \]

We see that \(Ax\) is a linear combination of the columns of the matrix \(A\), with the coefficients coming from the vector.

The first column of \(AB\) is equal to the product of \(A\) and the first column of \(B\):

\[ b_{11}C_{1}+b_{21}C_{2} \]

The second column of \(AB\) is equal to the product of \(A\) and the second column of \(B\):

\[ b_{12}C_{1}+b_{22}C_{2} \]

Remark: The product of a matrix and a vector is a linear combination of the columns. This is a very important result and will make an appearance throughout the course.