Note that the product of the two given matrices AB is not equal to BA. That is matrix
multiplication is not commutative. Matrix multiplication is commutative only when
the two given matrices are equal.
- Transpose of a Matrix
If A is the given matrix then its transpose denoted as AT is the matrix obtained by
interchanging the rows and columns of A. That is, to get the transpose of A, the rows
of the matrix A are to be interchanged as columns and the columns are to be
interchanged as rows.
If the order of matrix A is 3×4, then the order of the transpose matrix denoted as AT
is 4×3.
Note that in the case of a square matrix the order of a particular matrix and the order
of its transpose matrix remain the same.
The idea of transpose of a matrix can be taught to the child through real life experience.
For instance, in the classroom itself, a matrix with the students as the elements of
some order can be formed, and then the rows and columns of students can be
interchanged to form the transpose of the original matrix.
The idea can also be taught through the abacus. The usual abacus can form a matrix
of order 4×15 (or 4×13 as the case may be) in the lower abacus horizontally, and the
abacus when kept vertically will form a matrix of order 15×4, which is nothing but
the transpose of the original matrix formed in the lower abacus.
Eg : Let, A = ⎟⎟
⎠
⎞
⎜⎜⎝
⎛
−
− −
0 4 5
1 2 3
Therefore, the transpose matrix of A denoted as AT is given by,
AT = ⎟⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
− −
−
3 5
2 4
1 0
