If A=[ 1 4 2 5 3 6 ] and B=[ 1 2 3 4 5 6 ]

step 1 So in the given question we have a matrix A and B. The matrix A is given as 142536 and matrix B

If A=[ 1 4 2 5 3 6 ] and B=[ ...

If $A=\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right]$ and $B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]$, then (a) $\mathrm{AB}$ and $\mathrm{BA}$ do not exist (b) AB exists but BA does not exist (c) $\mathrm{AB}$ and $\mathrm{BA}$ exist and are equal (d) $\mathrm{AB}$ and $\mathrm{BA}$ exist but not equal

If $A=\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right]$ and $B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]$, then
(a) $\mathrm{AB}$ and $\mathrm{BA}$ do not exist
(b) AB exists but BA does not exist
(c) $\mathrm{AB}$ and $\mathrm{BA}$ exist and are equal
(d) $\mathrm{AB}$ and $\mathrm{BA}$ exist but not equal

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Key Concepts

Non-Commutativity of Matrix Multiplication

Matrix multiplication is generally non-commutative, meaning that AB does not necessarily equal BA, even when both products are defined. This property is significant in linear algebra as it affects how matrices are manipulated and helps in understanding the structure and behavior of different linear transformations.

Matrix Dimensions

Matrix dimensions define the size of a matrix as the number of rows and columns it contains. They are crucial when performing operations such as multiplication, as the product AB is defined only if the number of columns in matrix A equals the number of rows in matrix B. This concept helps in determining the feasibility of multiplying two matrices and predicting the dimensions of the resulting product.

Matrix Multiplication

Matrix multiplication is a binary operation that takes two matrices and produces another matrix, provided the dimensions are compatible. In this operation, each entry of the resulting matrix is computed as the sum of products of corresponding entries from a row of the first matrix and a column of the second matrix. Understanding this process ensures that one can correctly calculate the product when the dimensions allow it.

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