Find A B and B A in each case. LetA =[ 2 -3 1 ], B=[ 2 3 4 ], C=[

step 1 Okay, we want to find the product of B, E, with the first matrix B being 2, 3, 4, it's a 1 by 3

Find A B and B A in each case. LetA =[ 2 -3 ...

Find $A B$ and $B A$ in each case. $$\text { LetA }=\left[\begin{array}{r}2 \\-3 \\1\end{array}\right], B=\left[\begin{array}{lll} 2 & 3 & 4\end{array}\right], C=\left[\begin{array}{ll}2 & 3 \\4 & 5 \\1 & 0\end{array}\right]$$ $$D=\left[\begin{array}{rrr}2 & -1 & 1 \\0 & 3 & 2\end{array}\right], \text { and } E=\left[\begin{array}{lll}1 & 1 & 1 \\0 & 1 & 1 \\0 & 0 & 1\end{array}\right]$$ Find the following if possible. $$B E$$

Find $A B$ and $B A$ in each case.
$$\text { LetA }=\left[\begin{array}{r}2 \\-3 \\1\end{array}\right], B=\left[\begin{array}{lll}
2 & 3 & 4\end{array}\right], C=\left[\begin{array}{ll}2 & 3 \\4 & 5 \\1 & 0\end{array}\right]$$
$$D=\left[\begin{array}{rrr}2 & -1 & 1 \\0 & 3 & 2\end{array}\right], \text { and } E=\left[\begin{array}{lll}1 & 1 & 1 \\0 & 1 & 1 \\0 & 0 & 1\end{array}\right]$$
Find the following if possible.
$$B E$$

Key Concepts

Outer Product

When a column vector and a row vector are multiplied, the result is called the outer product. This operation yields a matrix where each element is the product of an entry from the column vector and an entry from the row vector, illustrating a specific case of matrix multiplication.

Non-Commutativity of Multiplication

Matrix multiplication is generally non-commutative, meaning that even if both products AB and BA are defined, they will not necessarily be equal. This is an important factor when evaluating different orders of multiplication.

Special Matrices and Their Products

Certain matrices, such as triangular matrices, have unique properties that can simplify multiplication processes or affect the resulting product. Recognizing these properties can assist in streamlining computations and understanding the structure of the resulting matrix.

Matrix Multiplication

Matrix multiplication is a binary operation that produces a matrix from two matrices. The process involves computing the dot product of rows from the first matrix with columns from the second matrix. It requires that the number of columns in the first matrix equals the number of rows in the second matrix, otherwise the product is not defined.

Conformability of Matrices

For two matrices to be multipliable, the inner dimensions must agree. Specifically, an m×n matrix can be multiplied by an n×p matrix, yielding an m×p result. This concept is crucial to determining whether operations such as AB, BA, or BE are possible.

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