Question

Let A be a 5 by 7, B be a 7 by 6 and C be a 6 by 5 matrix. How to determine the size of the following matrices? AB, BA, ${A^T}B$, BC, ABC, CA, ${B^T}A$, $B{C^T}$.

Answer 1

We know that the matrix is denoted by rows $\times$ columns and the size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an $m \times n$matrix, while m and n are called its dimensions, hence here we need to determine the size of the given matrix, in which A, B and C are the matrix defined with rows and columns, hence by theory of matrix multiplication we can find out the size of the matrices.

Complete step by step solution:
Let us write the given data
A, B and C are the defined matrix:
${A_{5 \times 7}}$, ${B_{7 \times 6}}$, ${C_{6 \times 5}}$
Hence, we need to determine the size of the following matrix:
AB, BA, ${A^T}B$, BC, ABC, CA, ${B^T}A$, $B{C^T}$
According to the theory of matrix multiplication, the matrix multiplication is only defined if B has the same number of columns as rows in A, if
${A_{m \times n}}$and ${B_{n \times p}}$, i.e.,
A has n columns and B has n rows, otherwise AB will not be defined, if it is defined as above, then the matrix AB will have m rows and p columns, i.e.,
${\left( {AB} \right)_{m \times p}}$
Furthermore, the transpose of a matrix is when rows become columns and columns become rows.
Hence, if
${A_{m \times n}} \Rightarrow {\left( {{A^T}} \right)_{n \times m}}$
Hence, in this given question we have:
${A_{5 \times 7}}$, ${B_{7 \times 6}}$, ${C_{6 \times 5}}$
Therefore, with respect to the above-mentioned theory, the size of AB is
${\left( {AB} \right)_{5 \times 6}}$
And BA is undefined,
${A^T}B$is undefined
$B{C_{7 \times 5}}$
Since matrix multiplication is associative,
$ABC = \left( {AB} \right)C = A\left( {BC} \right)$
Hence,
${\left( {ABC} \right)_{5 \times 5}}$
${\left( {CA} \right)_{6 \times 7}}$
${B^T}A$ is undefined.
$B{C^T}$ is undefined.

Note:
To solve any question on matrix we must note that, to multiply the matrix the number of columns of the first matrix must equal the number of rows of the second matrix and the result will have the same number of rows as the first matrix, and the same number of columns as the second matrix, hence to find the size of the matrix we must know the theory of multiplication as we have explained while solving.