Show that if A=[ a11 a12 a13 a21 a22 a23 ] and B=[ b11 b12

step 1 In this question we are given matrix is equal to 1 2 3 4 and matrix b is equal to 2 1 4 2 and ma

Show that if A=[ a11 a12 a13 a21 a22 ...

Show that if $$ A=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{llll} b_{11} & b_{12} & b_{13} & b_{14} \\ b_{21} & b_{22} & b_{23} & b_{24} \\ b_{31} & b_{32} & b_{33} & b_{34} \end{array}\right] \text {, } $$ then $$ A B=\left[\begin{array}{lll} a_{11} & 0 & 0 \\ a_{21} & 0 & 0 \end{array}\right] B+\left[\begin{array}{lll} 0 & a_{12} & 0 \\ 0 & a_{22} & 0 \end{array}\right] B+\left[\begin{array}{lll} 0 & 0 & a_{13} \\ 0 & 0 & a_{23} \end{array}\right] B $$ and hence that $$ A B=\left[\begin{array}{l} a_{11} \\ a_{21} \end{array}\right]\left[\begin{array}{lll} b_{11} & \cdots & b_{14} \end{array}\right]+\left[\begin{array}{l} a_{12} \\ a_{22} \end{array}\right]\left[\begin{array}{llll} b_{21} & b_{22} & b_{23} & b_{24} \end{array}\right]+\left[\begin{array}{l} a_{13} \\ a_{23} \end{array}\right]\left[\begin{array}{lll} b_{31} & \cdots & b_{34} \end{array}\right] \text {. } $$

Show that if
$$
A=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{llll}
b_{11} & b_{12} & b_{13} & b_{14} \\
b_{21} & b_{22} & b_{23} & b_{24} \\
b_{31} & b_{32} & b_{33} & b_{34}
\end{array}\right] \text {, }
$$
then
$$
A B=\left[\begin{array}{lll}
a_{11} & 0 & 0 \\
a_{21} & 0 & 0
\end{array}\right] B+\left[\begin{array}{lll}
0 & a_{12} & 0 \\
0 & a_{22} & 0
\end{array}\right] B+\left[\begin{array}{lll}
0 & 0 & a_{13} \\
0 & 0 & a_{23}
\end{array}\right] B
$$
and hence that
$$
A B=\left[\begin{array}{l}
a_{11} \\
a_{21}
\end{array}\right]\left[\begin{array}{lll}
b_{11} & \cdots & b_{14}
\end{array}\right]+\left[\begin{array}{l}
a_{12} \\
a_{22}
\end{array}\right]\left[\begin{array}{llll}
b_{21} & b_{22} & b_{23} & b_{24}
\end{array}\right]+\left[\begin{array}{l}
a_{13} \\
a_{23}
\end{array}\right]\left[\begin{array}{lll}
b_{31} & \cdots & b_{34}
\end{array}\right] \text {. }
$$

Key Concepts

Outer Product Decomposition

Outer product decomposition refers to expressing a matrix as a sum of rank-one matrices, each being the product of a column vector and a row vector. In the given problem, the product AB is written as the sum of three outer products: each term is a column from A multiplied by the corresponding row from B. This insight underpins many techniques in matrix analysis, including simplifying computations and understanding the structure and rank of matrices.

Matrix Multiplication

Matrix multiplication involves combining rows of the first matrix with columns of the second. In this problem, the product of a 2×3 and a 3×4 matrix is computed by summing the contributions of each of the matching pairs of entries from the row of the first matrix and the column of the second. This is a fundamental procedure in linear algebra and serves as the basis for many transformations and operations in higher-dimensional spaces.

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